Question

Which of the following is True about the function $ f(x) = x^4 - 4x^2 $ ?

• A. It has two local minima and one local maxima
• B. It has two local minima and zero local maxima
• C. It has one local minima and one local maxima
• D. It has two local minima and two local maxima

Answer 1

Answer: (A)

We get , $f \left(x\right)=x^{4}-4x^{2}$
$\Rightarrow f'\left(x\right)=4x^{3}-8x$
$=4x\left(x^{2}-2\right)$
$\Rightarrow f'' \left(x\right)=12x^{2}-8$
For maxima or minima, $f'\left(x\right) =0 $
$\Rightarrow 4x\left(x^{2}-2\right)=0$
$\Rightarrow x=0, \pm\sqrt{2}$
Now, $\left[f''\left(x\right)\right]_{x=0}=-8<\,0$
$\left[f''\left(x\right)\right]_{x=-\sqrt{2}} =24-8=16>\,0$
and $\left[f''\left(x\right)\right]_{x=-\sqrt{2}}=24-8=16>\,0$
Thus, $x = 0$ is a point of maxima and $x=\sqrt{2}, -\sqrt{2}$ are points of minima