Question:
Let $[y]$ represent the greatest integer less than or equal to $y$. Then the set of all $x$ at which $f(x)=\cos^{-1}[4x+3]$ is differentiable is
Let $[y]$ represent the greatest integer less than or equal to $y$. Then the set of all $x$ at which $f(x)=\cos^{-1}[4x+3]$ is differentiable is
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Functions inside greatest integer brackets are constant over step-intervals; differentiability fails at the transition boundary points where values jump.
Updated On: Jun 3, 2026
- $\mathbb{R}$
- $[-1,1]$
- $[-1,-\frac{1}{4})-\{-\frac{3}{4},-\frac{1}{2}\}$
- $(-\frac{3}{4},\infty)$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
The domain of $\cos^{-1}(u)$ is restricted to $-1 \le u \le 1$. Since $u = [4x+3]$ outputs only integers, $[4x+3]$ can only take values from the discrete integer set $\{-1, 0, 1\}$.
Step 2: Meaning
We break down the range values of the greatest integer function: 1. If $[4x+3] = -1 \implies -1 \le 4x+3 < 0 \implies -4 \le 4x < -3 \implies -1 \le x < -\frac{3}{4}$. Here, $f(x) = \cos^{-1}(-1) = \pi$. 2. If $[4x+3] = 0 \implies 0 \le 4x+3 < 1 \implies -3 \le 4x < -2 \implies -\frac{3}{4} \le x < -\frac{1}{2}$. Here, $f(x) = \cos^{-1}(0) = \frac{\pi}{2}$. 3. If $[4x+3] = 1 \implies 1 \le 4x+3 < 2 \implies -2 \le 4x < -1 \implies -\frac{1}{2} \le x < -\frac{1}{4}$. Here, $f(x) = \cos^{-1}(1) = 0$.
Step 3: Analysis
The function is a piecewise constant function on these sub-intervals. A piecewise constant function is differentiable with a derivative of 0 inside open intervals, but it is completely discontinuous at the integer transition boundary break-points: $x = -\frac{3}{4}$ and $x = -\frac{1}{2}$.
Step 4: Conclusion
Combining the safe open sub-intervals while removing the jump discontinuity points inside the outer bounds $[-1, -\frac{1}{4})$ yields the set of differentiability: $[-1,-\frac{1}{4})-\{-\frac{3}{4},-\frac{1}{2}\}$, matching option (C).
Final Answer: (C)
The domain of $\cos^{-1}(u)$ is restricted to $-1 \le u \le 1$. Since $u = [4x+3]$ outputs only integers, $[4x+3]$ can only take values from the discrete integer set $\{-1, 0, 1\}$.
Step 2: Meaning
We break down the range values of the greatest integer function: 1. If $[4x+3] = -1 \implies -1 \le 4x+3 < 0 \implies -4 \le 4x < -3 \implies -1 \le x < -\frac{3}{4}$. Here, $f(x) = \cos^{-1}(-1) = \pi$. 2. If $[4x+3] = 0 \implies 0 \le 4x+3 < 1 \implies -3 \le 4x < -2 \implies -\frac{3}{4} \le x < -\frac{1}{2}$. Here, $f(x) = \cos^{-1}(0) = \frac{\pi}{2}$. 3. If $[4x+3] = 1 \implies 1 \le 4x+3 < 2 \implies -2 \le 4x < -1 \implies -\frac{1}{2} \le x < -\frac{1}{4}$. Here, $f(x) = \cos^{-1}(1) = 0$.
Step 3: Analysis
The function is a piecewise constant function on these sub-intervals. A piecewise constant function is differentiable with a derivative of 0 inside open intervals, but it is completely discontinuous at the integer transition boundary break-points: $x = -\frac{3}{4}$ and $x = -\frac{1}{2}$.
Step 4: Conclusion
Combining the safe open sub-intervals while removing the jump discontinuity points inside the outer bounds $[-1, -\frac{1}{4})$ yields the set of differentiability: $[-1,-\frac{1}{4})-\{-\frac{3}{4},-\frac{1}{2}\}$, matching option (C).
Final Answer: (C)
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