Question

True or false: If a function is positive over an interval, then its derivative is also positive. Is this correct? Please explain to help me with a graphical example

Answer 1

The derivative only positive if it is increasing over an interval. It does not determine if the function is positive on that interval.

Given a sample function f(x)

`f(x)=x^2-8x+16`

We have the following graph

On the interval [0,4], the function f(x) is positive, however since the function is decreasing on that interval, its derivative is negative.

Therefore, it is false that when a derivative is positive then a function is positive over an interval.