Let f : [0, 4π] → [0, π] be defined by...

MathematicsProperties of Inverse Trigonometric FunctionsJEE Advanced 2014Moderate

Let $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = cos^{- 1} (cos x)$ . The number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ is

Answer:3

Visualized Solution (English)

Understanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (x) = cos^{- 1} (cos x)

Given function: $f (x) = cos^{- 1} (cos x)$
Domain: $x \in [0, 4 π]$ and Range: $y \in [0, π]$
The function is periodic with a period of $2 π$ .

Graphing the Sawtooth Wave

For $x \in [0, π]$ , $f (x) = x$
For $x \in [π, 2 π]$ , $f (x) = 2 π - x$
For $x \in [2 π, 3 π]$ , $f (x) = x - 2 π$
For $x \in [3 π, 4 π]$ , $f (x) = 4 π - x$

The Linear Equation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y = 1 - \frac{x}{10}

Equation to solve: $f (x) = \frac{10 - x}{10}$
Let $g (x) = 1 - \frac{x}{10}$
At $x = 0, g (0) = 1$
At $x = 10, g (10) = 0$

Intersections in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [0, 2 π]

In $[0, π]$ : $x = 1 - \frac{x}{10} ⟹ 1.1 x = 1 ⟹ x \approx 0.91$ (Point 1)
In $[π, 2 π]$ : $2 π - x = 1 - \frac{x}{10} ⟹ 0.9 x = 2 π - 1 ⟹ x \approx 5.87$ (Point 2)

Intersections in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [2 π, 4 π]

In $[2 π, 3 π]$ : $x - 2 π = 1 - \frac{x}{10} ⟹ 1.1 x = 1 + 2 π ⟹ x \approx 6.62$ (Point 3)
In $[3 π, 4 π]$ : $4 π - x = 1 - \frac{x}{10} ⟹ 0.9 x = 4 π - 1 ⟹ x \approx 12.85$
Since $12.85 > 4 π$ , there is no intersection in the last segment.

Final Count of Solutions

Total number of intersection points = 3
The intersections occur at $x \approx 0.91, 5.87, 6.62$
Final Answer: 3

00:00 / 00:00

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Let $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = cos^{- 1} (cos x)$ . The number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ is

Visualized Solution (English)

Understanding $f (x) = cos^{- 1} (cos x)$

Graphing the Sawtooth Wave

The Linear Equation $y = 1 - \frac{x}{10}$

Intersections in $[0, 2 π]$

Intersections in $[2 π, 4 π]$

Final Count of Solutions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[0,4π]→[0,π] be defined by f(x)=cos−1(cosx). The number of points x∈[0,4π] satisfying the equation f(x)=1010−x​ is

Visualized Solution (English)

Understanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)=cos−1(cosx)

Graphing the Sawtooth Wave

The Linear Equation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y=1−10x​

Intersections in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [0,2π]

Intersections in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [2π,4π]

Final Count of Solutions

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