Let x=2 be a local minima of the function [f(x)=2x^4-18x^2+8x+12,x(-4,4)].

Question

Let $x=2$ be a local minima of the function $$f(x)=2x^4-18x^2+8x+12,x\in(-4,4)$$. If M is local maximum value of the function $f$ in ($-4,4)$, then M =

Accepted Answer

$$
\begin{aligned}
& f(x)=8 x^3-36 x+8 \\
& =4\left(2 x^3-9 x+2\right) \\
& =4(x-2)\left(2 x^2+4 x-1\right) \\
& =4(x-2)\left(x-\frac{-2+\sqrt{6}}{2}\right)\left(x-\frac{-2 \sqrt{6}}{2}\right)
\end{aligned}
$$

Local maxima occurs at $x=\frac{-2+\sqrt{6}}{2}=x_0$

$$
f\left(x_0\right)=12 \sqrt{6}-\frac{33}{2}
$$

Solutions · 1