Question

For the function $f\left(x\right)=\left(sin\right)^{3} x-3sin⁡x+4 \, \forall x\in \left[0 , \frac{\pi }{2}\right],$ which of the following is true?

• A. Greatest value of the function is $2\pi $
• B. Greatest value of the function is $4$
• C. Rolle’s Theorem is applicable to $f\left(x\right)$ in $x\in \left[0 , \frac{\pi }{2}\right]$
• D. LMVT is not applicable to $f\left(x\right)$ in $x\in \left[0 , \frac{\pi }{2}\right]$

Answer 1

Answer: (B)

Let, $\sin x=t \in[0,1]$
$\therefore f(t)=t^{3}-3 t+4$
Now, $f^{\prime}(t)=3 t^{2}-3$
$=3(t-1)(t+1)$
$\therefore f^{\prime}(t) \leq 0 \forall t \in[0,1]$
$\therefore \max (f(t))=0-0+4=4$
Also, $f(x)$ is continuous in [0,1] and differentiable in $(0,1),$ hence LMVT
is applicable. But, $f(0) \neq f(1),$ hence Rolle's theorem is not applicable