Question

How do you find the local max and min for `f(x)=2x^3 + 5x^2 - 4x - 3`?

Answer 1

`x=-2` is a local maximum, and `x=1/3` is a local minimum.
See explanations below.

Explanation:

`f(x)=2x³+5x²-4x-3`
What we know is that there's a local extremum when `f'(x)=0`
`f'(x)=6x²+10x-4`
`6x²+10x-4=0`
`6(x²+5/3x-2/3)=0`
`x²+5/3x-2/3=0`
`x²-x/3+2x-2/3=0`
`x(x-1/3)+2(x-1/3)=0`
`(x+2)(x-1/3)=0`
Now we can clearly see that when `x=-2` and `x=1/3`, `f'(x)=0`
Also, we know that out of roots, `f(x)=ax³+bx²+cx+d` take the sign of `-a` in `-oo` and the sign of `a` in `+oo`. Because of that, we can deduce that `x=-2` is a local maximum, and `x=1/3` is a local minimum.
\0/ here's our answer!