Question

At what points on the graph of `f(x)=2x^3-9x^2-12x+5` is the slope of the tangent line 12?

Answer 1

The derivative of a function gives us the slope of a tangent line for a specified value of `x`

`f(x) = 2x^3-9x^2-12x+5`
implies
`f'(x) = 6x^2-18x-12`

and we are asked to find when this is `=12`

`6x^2-18x-12 = 12`

`x^2-3x -4=0`

which factors as
`(x-4)(x+1)=0`

when `x=4`
`f(x) = 2(4)^3-9(4)^2-12(4)+5`
`=-59`

when `x=-1`
`f(-1)=2(-1)^3-9(-1)^2-12(-1)+5`
`=6`

So `f(x)` has a tangent with slope `12` at
`(4,-59)`
and
`(-1,6)`