Question

If $f\left(x\right)=\cos \left\{\frac{\mathrm{\pi }}{2}\left[x\right]-x^3\right\},1<x<2$ and $\left[x\right]$ is the greatest integer function, then $f\text{'}\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)$ is equal to

• A. $0$
• B. ${\left(\frac{\mathrm{\pi }}{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$
• C. $3{\left(\frac{\mathrm{\pi }}{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$
• D. $-3{\left(\frac{\mathrm{\pi }}{2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Answer 1

The correct option is A

$0$

Explanation for the correct option.

Step 1. Simplify the given function.

For $1<x<2$, the value of greatest integer function $\left[x\right]$ is equal to $1$.

So in the given function $f\left(x\right)=\cos \left\{\frac{\mathrm{\pi }}{2}\left[x\right]-x^3\right\}$, substitute $1$ for $\left[x\right]$ and simplify.

$\begin{array}{rcl}f\left(x\right)& =& \cos \left\{\frac{\mathrm{\pi }}{2}\times 1-x^3\right\}\\ & =& \cos \left(\frac{\mathrm{\pi }}{2}-x^3\right)\\ & =& \sin \left(x^3\right)\left[\cos \left(\frac{\mathrm{\pi }}{2}-\theta \right)=\sin \theta \right]\end{array}$

Step 2. Find the value of $\begin{array}{rcl}& & f\text{'}\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)\end{array}$.

Differentiate the function $f\left(x\right)=\sin \left(x^3\right)$ .

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(\sin \left(x^3\right)\right)\\ & =& \cos \left(x^3\right)\times 3x^2\\ & =& 3x^2\cos \left(x^3\right)\end{array}$

Now the value of $\begin{array}{l}f\text{'}\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)\end{array}$ is given as

$\begin{array}{rcl}f\text{'}\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)& =& 3{\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)}^2\cos \left({\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)}^3\right)\\ & =& 3{\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)}^2\cos \left(\frac{\mathrm{\pi }}{2}\right)\\ & =& 3{\left(\sqrt[3]{\frac{\mathrm{\pi }}{2}}\right)}^2\times 0\\ & =& 0\end{array}$

Hence, the correct option is A.