Question

If f(x) = cos $\left\{ \frac{\pi}{2}\lbrack x\rbrack - x^{3} \right\}$, – 1 < x < 2 and [x] is the greatest integer less than or equal to x then f ' $\left( \sqrt[3]{\frac{\pi}{2}} \right)$ is –

• A. 0
• B. 1
• C. ½
• D. 1/$\sqrt{2}$

Answer 1

Answer: (A)

0

Answer 2

We know

1 < $\sqrt[3]{\frac{\pi}{2}}$ < 2

$\left\lbrack \sqrt[3]{\frac{\pi}{2}} \right\rbrack$ = 1

Now f(x) = cos $\left\{ \frac{\pi}{2}\lbrack x\rbrack - x^{3} \right\}$

f$\left( \sqrt[3]{\frac{\pi}{2}} \right)$ = cos $\left\{ \frac{\pi}{2}\left\lbrack \sqrt[3]{\frac{\pi}{2}} \right\rbrack - \left( \sqrt[3]{\frac{\pi}{2}} \right)^{3} \right\}$

= cos $\left\{ \frac{\pi}{2} - \frac{\pi}{2} \right\}$

= cos 0 = 1

f ' $\left( \sqrt[3]{\frac{\pi}{2}} \right)$ = 0