Question

Find the points of maxima, minima and the intervals of monotonicity of the following function:
$y=4x^4-2x^2+3$

Answer 1

$y=4x^4-2x^2+3$

$\frac{dy}{dx}=16x^3-4x$

for increasing $\frac{dy}{dx}>0$

$\Rightarrow 16x^3-4x>0$

$\Rightarrow 4x(4x^2-1)>0$

$\Rightarrow 4x(2x+1)(2x-1)>0$

$xϵ(-\frac{1}{2},0)\cup (\frac{1}{2},\infty )$

and for decreasing $f^{\prime }\left(x\right)<0$

so $xϵ(-\infty ,-\frac{1}{2})\cup (0,\frac{1}{2})$

and for maxima and minima $f^{\prime }\left(x\right)=0$

$4x^2(4x^2-1)=0$

$x=0,x=\frac{1}{2},-\frac{1}{2}$

$f^{\prime \prime }\left(x\right)=48x^2-4$

$f^{\prime \prime }\left(0\right)=4$ (point of local maxima)

$f^{\prime \prime }\left(\frac{1}{2}\right)=48\times \frac{1}{4}-4$

$=8$ (point of local minima)

$f^{\prime \prime }\left(\frac{-1}{2}\right)=48\times \frac{1}{4}-4$

$=8$ (point of local minima).