Let f(x) = (x-1)² sin(1/(x-1)) - |x| if...

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1981Moderate

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Answer:

{0}

Visualized Solution (English)

Identifying Potential Trouble Points

Given function: $f (x) = {(x - 1)^{2} sin \frac{1}{x - 1} - ∣ x ∣ - 1 x \neq = 1 x = 1$
Potential points of non-differentiability: $x = 1$ (definition change) and $x = 0$ (due to $∣ x ∣$ ).

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = 1

Check continuity at $x = 1$ :
$lim_{x \to 1} f (x) = lim_{x \to 1} [(x - 1)^{2} sin \frac{1}{x - 1} - ∣ x ∣]$
Using Squeeze Theorem: $lim_{x \to 1} (x - 1)^{2} sin \frac{1}{x - 1} = 0$
$lim_{x \to 1} f (x) = 0 - ∣1∣ = - 1$
Since $lim_{x \to 1} f (x) = f (1)$ , the function is continuous at $x = 1$ .

Checking Differentiability at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = 1

Derivative at $x = 1$ using first principles:
$f^{'} (1) = lim_{h \to 0} \frac{f ( 1 + h ) - f ( 1 )}{h}$
$f^{'} (1) = lim_{h \to 0} \frac{[ h ^{2} s i n ( 1/ h ) - ∣1 + h ∣ ] - ( - 1 )}{h}$
For small $h$ , $∣1 + h ∣ = 1 + h$ :
$f^{'} (1) = lim_{h \to 0} \frac{h ^{2} s i n ( 1/ h ) - 1 - h + 1}{h} = lim_{h \to 0} (h sin \frac{1}{h} - 1)$
$f^{'} (1) = 0 - 1 = - 1$ . So, $f (x)$ is differentiable at $x = 1$ .

Analyzing the Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = 0

At $x = 0$ :
Part 1: $g (x) = (x - 1)^{2} sin \frac{1}{x - 1}$ is differentiable at $x = 0$ .
Part 2: $h (x) = ∣ x ∣$ is not differentiable at $x = 0$ .
Property: Differentiable function $\pm$ Non-differentiable function $=$ Non-differentiable function.
Therefore, $f (x)$ is not differentiable at $x = 0$ .

Final Conclusion

The function $f (x)$ is differentiable everywhere except at $x = 0$ .
Final Set of points: ${0}$
Key Takeaway: Damping terms like $(x - a)^{n}$ can make oscillating functions differentiable at $x = a$ if $n > 1$ .

00:00 / 00:00

Conceptually Similar Problems

MathematicsDifferentiability of a FunctionJEE Main 2008Moderate

View in:English Hindi

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

(A)

f is neither differentiable at x = 0 nor at x = 1

(B)

f is differentiable at x = 0 and at x = 1

(C)

f is differentiable at x = 0 but not at x = 1

(D)

f is differentiable at x = 1 but not at x = 0

Answer:C

MathematicsDifferentiability of a FunctionJEE Advanced 2005Moderate

View in:English Hindi

The function given by $y = ∣∣ x ∣ - 1∣$ is differentiable for all real numbers except the points

(A)

{0, 1, - 1}

(B)

\pm 1

(C)

(D)

- 1

Answer:A

MathematicsDifferentiability of a FunctionJEE Advanced 1999Moderate

View in:English Hindi

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

(A)

- 1

(B)

(C)

(D)

Answer:D

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1987Easy

View in:English Hindi

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

(A)

(- \infty, \infty)

(B)

[0, \infty)

(C)

(- \infty, 0) \cup (0, \infty)

(D)

(0, \infty)

(E)

None

Answer:A

MathematicsDifferentiability of a FunctionJEE Advanced 2001Moderate

View in:English Hindi

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

(A)

{- 1, 1}

(B)

{- 1, 0}

(C)

{0, 1}

(D)

{- 1, 0, 1}

Answer:D

MathematicsDifferentiability of a FunctionJEE Main 2006Easy

View in:English Hindi

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

(A)

(- \infty, 0) \cup (0, \infty)

(B)

(- \infty, - 1) \cup (- 1, \infty)

(C)

(- \infty, \infty)

(D)

(0, \infty)

Answer:C

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1986Moderate

View in:English Hindi

The function $f (x) = 1 + ∣ sin x ∣$ is

* Multiple Correct Options

(A)

continuous nowhere

(B)

continuous everywhere

(C)

differentiable nowhere

(D)

not differentiable at

x = 0

(E)

not differentiable at infinite number of points

Answer:B, D, E

MathematicsDifferentiability of a FunctionJEE Advanced 2012Moderate

View in:English Hindi

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

(A)

differentiable both at

x = 0

and at

x = 2

(B)

differentiable at

x = 0

but not differentiable at

x = 2

(C)

not differentiable at

x = 0

but differentiable at

x = 2

(D)

differentiable neither at

x = 0

nor at

x = 2

Answer:B

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 2011Moderate

View in:English Hindi

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

* Multiple Correct Options

(A)

f (x)

is continuous at

x = - \frac{π}{2}

(B)

f (x)

is not differentiable at

x = 0

(C)

f (x)

is differentiable at

x = 1

(D)

f (x)

is differentiable at

x = - \frac{3}{2}

Answer:A, B, C, D

MathematicsDifferentiability of a FunctionJEE Main 2007Moderate

View in:English Hindi

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

(A)

f(x) is differentiable everywhere

(B)

f(x) is not differentiable at x = 0

(C)

f(x) \geq 1 for all x \in R

(D)

f(x) is not differentiable at x = 1

Answer:A

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Visualized Solution (English)

Identifying Potential Trouble Points

Checking Continuity at $x = 1$

Checking Differentiability at $x = 1$

Analyzing the Point $x = 0$

Final Conclusion

Conceptually Similar Problems

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

The function given by $y = ∣∣ x ∣ - 1∣$ is differentiable for all real numbers except the points

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

The function $f (x) = 1 + ∣ sin x ∣$ is

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

The function given by $y = ∣∣ x ∣ - 1∣$ is differentiable for all real numbers except the points

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

The function $f (x) = 1 + ∣ sin x ∣$ is

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)2sin(x−1)1​−∣x∣−1​if x=1if x=1​ be a real-valued function. Then the set of points where f(x) is not differentiable is .........

Visualized Solution (English)

Identifying Potential Trouble Points

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=1

Checking Differentiability at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=1

Analyzing the Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=0

Final Conclusion

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)sinx−11​,0,​if x=1if x=1​. Then which one of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function given by y=∣∣x∣−1∣ is differentiable for all real numbers except the points

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=(x2−1)∣x2−3x+2∣+cos(∣x∣) is NOT differentiable at

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=max{x,x3}. The set of all points where f(x) is NOT differentiable is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of points where f(x)=1+∣x∣x​ is differentiable is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={x2cosxπ​0​x=0x=0​,x∈R then f is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)sinx−11​,0,​if x=1if x=1​. Then which one of the following is true?

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=min{x+1,∣x∣+1}. Then which of the following is true?

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Checking Continuity at $x = 1$

Checking Differentiability at $x = 1$

Analyzing the Point $x = 0$

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

The function given by $y = ∣∣ x ∣ - 1∣$ is differentiable for all real numbers except the points

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

The function $f (x) = 1 + ∣ sin x ∣$ is

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

The function given by $y = ∣∣ x ∣ - 1∣$ is differentiable for all real numbers except the points

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

The function $f (x) = 1 + ∣ sin x ∣$ is

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?