Question

Find the relative extrema for the following functions by (1) determining the critical value(s) and (2) determining whether the critical values of the function is a relative maximum or minimum (or possible inflection point) f(x) = 2x³ + 18x² + 36x + 20 Ox=-3-2√3 relative maximum; x = -3+2√3 relative minimum x = -3 relative maximum Ox=3-√3 relative maximum x = 3+√3 relative minimum None of the above

Answer 1

`x=-3+√(3)` is a relative minimum

`x=-3-√(3)` is a relative maximum

Explanation:

Determine critical values

`f(x)=2x^3+18x^2+36x+20\\f'(x)=6x^2+36x+36\\\\0=6x^2+36x+36\\0=x^2+6x+6\\0+3=x^2+6x+6+3\\3=x^2+6x+9\\3=(x+3)^2\\\pm√(3)=x+3\\x=-3\pm√(3)\\x\approx-1.268,-4.732`

Use test points

`f'(0)=6(0)^2+36(0)+36=36 > 0\\f'(-2)=6(-2)^2+36(-2)+36=-12 < 0\\f'(-5)=6(-5)^2+36(-5)+36=6 > 0`

Therefore,
`-3+√(3)` is a relative minimum and
`-3-√(3)` is a relative maximum