Question

The minimum value of the derivative of f(x)=3x^5+5x^4 occurs when x=

Answer 1

-5

Explanation:

Given
`f(x)=3x^(5)+5x^(4)`.

Derivative of
`f(x)` is given by
`(df(x))/(dx) =15x^(4)+20x^(3)=g(x)`.

Minimum value of
`g(x)` can be found differentiating again and equating it to zero.

`(dg(x))/(dx)=60x^(3)+60x^(2)=0\\\\x^(2)=0\text{ (or) }60x+60=0\\x=0\text{ (or) }x=-1`

Function
`g(x)` takes a minimum value when
`(d^(2)g(x))/(dx^(2))>0`

For,
`g''(x)=180x^(2)+120x`,
`g''(0)=0;\\g''(-1)=60>0`

So, minimum value occurs at
`x=-1`;

Minimum value =
`15(-1)^(4)+20(-1)^(3)=-5`

∴ Minimum value = -5