An insect is crawling along the curve $f(x)=\frac{x^4}{4}-\frac{8x^3}{3}+\frac{13}{2}x^2-6x+11.$ At what value of x, insect can attain its maximum or minimum value ?

x=6

The correct answer is Option (4) → $x=6$

$f(x)=\frac{x^4}{4}-\frac{8x^3}{3}+\frac{13}{2}x^2-6x+11$

for critical point, $f'(c)=0$

$⇒c^3-8c^2+13c-6=0$

$⇒(c-1)^2(c-6)=0$

∴ At $x=6$,$f(x)$ can achieve its maximum or minimum value.