Differentiation Rules for Exponential Functions

Question

Identify the coordinates of any local extrema of the function y= e^x-e^2x

Answer 1

Dear Shreya,

To answer this question, you have to first know what a local extreme (extremum, in Latin, I suppose) is: it's a place x where the function f(x) has some value, and immediately to each side (for x > or < that x value) of that has values that differ from f(x) in the same signed direction -- in other words, the function goes down on both sides, or the function goes up on both sides.

Since you are talking differentiation, you know that differentiating your function will allow you to find places where the slope is zero -- places which MIGHT be extrema, but aren't necessarily (they might also be places where the function is discontinuous, or merely has a zero-slope inflection point, among other possibilities).

However, if you take the second derivative at such places, and that has a smooth-functioned value different than zero (this is the curvature of the function), that will rule out inflection points, and you will know you have an extremum (plural = extrema). (Rarely, both first and second derivatives might be zero, and you'd then check the third derivative, etc. == L'Hospital's Rule.)

So, in your function f(x) = e^x - e^2x, f'(x) = e^x - 2e^x ; does this have any zeros? Only if e^x = 0. This doesn't happen except at x = - infinity, which is by definition NOT local.

So your answer should be, "There are no local extrema to the function" or "The empty set", or whatever.