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Step-by-Step Solution
Step 1: Understand the Function
We are given a function We want to find all points in the interval where this function is not differentiable.
Step 2: Reason for Non-Differentiability
The function is formed by taking the smaller of $\sin x$ and $\cos x$. Such a piecewise definition can lead to "corners" (sharp turns) at the boundary points where or where the expression for the minimum switches from $\sin x$ to $\cos x$ (or vice versa). At these boundaries, the left-hand derivative and right-hand derivative typically differ, making the function non-differentiable there.
Step 3: Identify the Regions Where One Function is Smaller
To determine the intervals over which $f(x) = \sin x$ versus $f(x) = \cos x$, we compare $\sin x$ and $\cos x$ in different subintervals of $(-\pi, \pi)$. The main switching points occur where sinx=cosx\sin x = \cos x and also where the sign of these functions might change. Let us locate these points precisely:
Solving $\sin x = \cos x$:
We have $\sin x - \cos x = 0 \implies \sin x = \cos x \implies \tan x = 1.$
In the interval $(-\pi, \pi)$, the solutions to $\tan x = 1$ are:
Checking other boundary points: Through quadrant-wise analysis of $\sin x$ and $\cos x$ (looking at points where each function might become negative or positive), we find that the function $f(x)$ also switches its definition around $x = -\frac{\pi}{4}$ and $x = \frac{3\pi}{4}.$
Hence, the function $f(x)$ changes from $\sin x$ to $\cos x$ or vice versa at the four points: At these points, the piecewise description of $f(x)$ changes and typically causes a corner (nondifferentiable point).
Step 4: Conclusion
Therefore, the set of all points in (-π,π)(-\pi, \pi) where f(x)=min{sinx,cosx}f(x) = \min \{\sin x, \cos x\} is not differentiable is precisely:
The correct choice given the options is
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Step-by-Step Solution
Step 1: Understand the Function
We are given a function f(x)=min{sinx,cosx}.f(x) = \min \{\sin x, \cos x\}. We want to find all points in the interval (-π,π)(-\pi, \pi) where this function is not differentiable.
Step 2: Reason for Non-Differentiability
The function f(x)=min{sinx,cosx}f(x) = \min \{\sin x, \cos x\} is formed by taking the smaller of $\sin x$ and $\cos x$. Such a piecewise definition can lead to "corners" (sharp turns) at the boundary points where sinx=cosx\sin x = \cos x or where the expression for the minimum switches from $\sin x$ to $\cos x$ (or vice versa). At these boundaries, the left-hand derivative and right-hand derivative typically differ, making the function non-differentiable there.
Step 3: Identify the Regions Where One Function is Smaller
To determine the intervals over which $f(x) = \sin x$ versus $f(x) = \cos x$, we compare $\sin x$ and $\cos x$ in different subintervals of $(-\pi, \pi)$. The main switching points occur where sinx=cosx\sin x = \cos x and also where the sign of these functions might change. Let us locate these points precisely:
Solving $\sin x = \cos x$:
We have $\sin x - \cos x = 0 \implies \sin x = \cos x \implies \tan x = 1.$
In the interval $(-\pi, \pi)$, the solutions to $\tan x = 1$ are: x=π4 and x=-3π4.x = \frac{\pi}{4} \quad\text{and}\quad x = -\frac{3\pi}{4}.
Checking other boundary points: Through quadrant-wise analysis of $\sin x$ and $\cos x$ (looking at points where each function might become negative or positive), we find that the function $f(x)$ also switches its definition around $x = -\frac{\pi}{4}$ and $x = \frac{3\pi}{4}.$
Hence, the function $f(x)$ changes from $\sin x$ to $\cos x$ or vice versa at the four points: -3π4, -π4, π4, 3π4. -\frac{3\pi}{4}, \quad -\frac{\pi}{4}, \quad \frac{\pi}{4}, \quad \frac{3\pi}{4}. At these points, the piecewise description of $f(x)$ changes and typically causes a corner (nondifferentiable point).
Step 4: Conclusion
Therefore, the set SS of all points in (-π,π)(-\pi, \pi) where f(x)=min{sinx,cosx}f(x) = \min \{\sin x, \cos x\} is not differentiable is precisely: -3π4, -π4, π4, 3π4. \left\{-\frac{3\pi}{4},\; -\frac{\pi}{4},\; \frac{\pi}{4},\; \frac{3\pi}{4}\right\}.
The correct choice given the options is -3π4,-π4,3π4,π4. \left\{ - \frac{3\pi}{4}, - \frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{4} \right\}.
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