# What are the local extrema of f(x)= x^3-7x?

Dec 23, 2015

Turning points (local extrema) occur when the derivative of the function is zero,
ie when $f ' \left(x\right) = 0$.
that is when $3 {x}^{2} - 7 = 0$
$\implies x = \pm \sqrt{\frac{7}{3}}$.

since the second derivative $f ' ' \left(x\right) = 6 x$, and
$f ' ' \left(\sqrt{\frac{7}{3}}\right) > 0 \mathmr{and} f ' ' \left(- \sqrt{\frac{7}{3}}\right) < 0$,

it implies that $\sqrt{\frac{7}{3}}$ is a relative minimum and $- \sqrt{\frac{7}{3}}$ is a relative maximum.

The corresponding y values may be found by substituting back into the original equation.

The graph of the function makes verifies the above calculations.

graph{x^3-7x [-16.01, 16.02, -8.01, 8]}