Does f Have a Local Extrema at x = 0?

Date: 07/10/2003 at 16:29:40
From: Sara
Subject: I'm horrible with trig

f(x) = (x^3)/6 + (x^2)/2 + cosx - 1

Does f have a local extremum at x = 0?

Date: 07/10/2003 at 17:54:01
From: Doctor Jaffee
Subject: Re: I'm horrible with trig

Hi Sara,

If a function is differentiable for all values of x (which your 
function is), all that you have to worry about is if the first 
derivative is equal to zero and the second derivative isn't.

The reason is that if the first derivative equals zero, then the 
tangent line is horizontal, so the point is either a maximum, a 
minimum, or possibly an inflection point. If the second derivative is 
zero, then the point is an inflection point.

So, calculate f'(0) and f"(0) and you should have the answer to your 
question.

Give it a try and if you want to check your solution with me or if you 
have difficulties or other questions write back to me and I'll try to 
help you some more.

Good luck,

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/