Question

For the function $f\left(x\right)={\left(sin\right)}^{3}x-3sinx+4\forall x\in \left[0,\frac{\pi }{2}\right],$ 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, $\sin x=t\in [0,1]$
$\therefore f(t)=t^3-3t+4$
Now, $f^{\prime }(t)=3t^2-3$
$=3(t-1)(t+1)$
$\therefore f^{\prime }(t)\le 0\forall t\in [0,1]$
$\therefore \max (f(t))=0-0+4=4$
Also, $f(x)$ is continuous in [0,1] and differentiable in $(0,1),$ hence LMVT
is applicable. But, $f(0)\ne f(1),$ hence Rolle's theorem is not applicable