Question

Find the intervals for f(x)$=\frac{x^4}{12}-\frac{5x^3}{6}+3x^2+7$ in which it is concave downward.

• A. $(2,3)$
• B. $(2,-3)$
• C. $(-2,2)$
• D. $(-3,3)$

Answer 1

The correct option is A $(2,3)$

Given, $f\left(x\right)=\frac{x^4}{12}-\frac{5x^3}{6}+3x^2+7$
$f^{\prime }\left(x\right)=\frac{x^3}{3}-\frac{5x^2}{2}+6x$
$f^{\prime \prime }\left(x\right)=x^2-5x+6=(x-2)(x-3)$
Now for $f\left(x\right)$ to be concave upward $f^{\prime \prime }\left(x\right)<0$

$\Rightarrow x\in (2,3)$