At which points does $f(x) = x^{4} - 8 x^{2} + 3$ have local minima?

x = 0

onlyCorrect answerYour answer

x = -2

and

x = 2

Correct answerYour answer

x = -2

x = 0

and

x = 2

Correct answerYour answer

x = 2

onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$f'(x) = 4x^3 - 16x = 4x(x - 2)(x + 2)$ , which is 0 at $-2$ , 0 and 2.
The sign of $f'$ runs $-, +, -, +$ across these points: it turns from negative to positive at $-2$ and 2 (minima) and from positive to negative at 0 (a maximum).

Worth noting

$x = 0$ is a critical point too, but there $f$ rises and then falls: a local maximum.

Ten questions from Maxima & Minima, at your own pace.

At which points does f(x)=x4−8x2+3f(x) = x^{4} - 8 x^{2} + 3f(x)=x4−8x2+3 have local minima?