T or F... If f is a positive function, then its derivative is also a positive function?

Question

T or F... If f is a positive function, then its derivative is also a positive function?

Answer 1

False.

Take f(x) = e^-x. This is always positive, but its derivative f'(x) = -e^-x is in fact always negative.

Answer 2

I'm not sure what a positive function is and I cannot find it in any of my math books or Wikipedia, so I'll guess that you mean a function that is always positive.

f(x) = 2 - sin(x) is a positive function

However f'(x) = cos(x)  and cos(x) is not always positive.  It varies from -1 to +1.

Answer 3

Derivative is like the slope of the function.

The derivative is positive if the function is slopping up.

The derivative is negative if the function is slopping down.

Sure, a function that is positive can slope up or slope down.