For $y=x^3\, log_ex, x> 0$

A. $\left.\frac{dy}{dx}\right]_{x=e}=4e$

B. $\frac{d^2y}{dx^2}=x^2(5+log_ex^3)$

C. $\frac{dy}{dx}=x^2(1+log_ex^3)$

D. $\left.\frac{d^2y}{dx^2}\right]_{x=e}=11e$

E. $\left.\frac{d^2y}{dx^2}\right]_{x=1}=5$

Choose the correct answer from the options given below :

C, D, E

The correct answer is Option (3) → C, D, E

$y=x^3\log_ex$

$\frac{dy}{dx}=3x^2\log_ex+x^2$

$=x^2(3\log_ex+1)=x^2(\log_ex^3+1)$ → (C)

Now,

$\frac{d^2y}{dx^2}=3(2x\log_ex+x)+2x$

$\left.\frac{d^2y}{dx^2}\right|_{x=e}=3(2e\log_ee+e)+2e$

$=3(2e+e)+2e$

$=11e$ → (D)

$\left.\frac{d^2y}{dx^2}\right|_{x=1}=3(2×0+1)+2(1)$

$=5$ → (E)