Question

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

Answer 1

Turning points (local extrema) occur when the derivative of the function is zero,
ie when `f'(x)=0`.
that is when `3x^2-7=0`
`=>x=+-sqrt(7/3)`.

since the second derivative `f''(x)=6x`, and
`f''(sqrt(7/3))>0 and f''(-sqrt(7/3))<0`,

it implies that `sqrt(7/3)` is a relative minimum and `-sqrt(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]}