Question

How do you find the local maximum and minimum values of `f(x) = 2x^3 - 5x +1` in the the interval is (-3,3)?

Answer 1

`x=-sqrt(5/6)` is a local maximum
`x=sqrt(5/6)` is a local minimum
graph{2x^3-5x+1 [-10, 10, -5, 5]}

Explanation:

Find local extrema on the interval by finding where `f'(x)` is equal to zero. First find `f'(x)`

`f(x)=2x^3-5x+1`

`f'(x)=6x^2-5`
`0=6x^2-5`
`5=6x^2`
`x^2=5/6`
`x=+-sqrt(5/6)`
`xapprox+-.9129`

Find whether each is a local max or local min by checking values around `+-sqrt(5/6)`;
`x=-sqrt(5/6)` is a local maximum
`x=sqrt(5/6)` is a local minimum