Question

How do you find the exact relative maximum and minimum of the polynomial function of `f(x) = x^3 + 4x^2 - 5x`?

Answer 1

minimum: `~~(0.523,-1.378)`

maximum: `~~(-3.189,24.193)`

Explanation:

relative max/min can only exist if `f'(x)` is 0 at that x-value.

first find `f'(x)`:
`=3x^2+8x-5` (power rule)

solving this quadratic: `x=(-8+-sqrt(8^2-4(3)(-5)))/(2*3)`
`x=(-8+-sqrt(124))/6`
`x=(-8+-2sqrt(31))/6`
`x=(-4+-sqrt(31))/3`

since `f(x)` is a cubic with a positive leading coefficient, it will approach `-oo` on the left and `+oo` on the right. this means the lesser x-value where `f'(x)=0` , (`(-4-sqrt(31))/3`) , is a local maximum and the greater x-value , (`(-4+sqrt(31))/3`) , is a local minimum.

plugging the x-values into `f(x)` gives:
minimum: `(-4/3+sqrt(31)/3,308/27-(62sqrt(31))/27)` `~~(0.523,-1.378)`

maximum: `(-4/3-sqrt(31)/3,308/27+(62sqrt(31))/27)` `~~(-3.189,24.193)`