Finding Local Extrema Analytically

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• Jan 10th 2008, 10:07 PM
Truthbetold
Finding Local Extrema Analytically
How do you do it?
Say the problem $\displaystyle \frac{1}[x^2}$.
The critical points are only a few.
How do you figure out whether this is the absolute or local without looking at the graph?

Thank you.
• Jan 10th 2008, 10:14 PM
Jhevon
Quote:

Originally Posted by Truthbetold
How do you do it?
Say the problem $\displaystyle \frac{1}[x^2}$.
The critical points are only a few.
How do you figure out whether this is the absolute or local without looking at the graph?

Thank you.

$\displaystyle \frac 1{x^2}$ has no critical points...er, unless you count where it's undefined as a critical point, which would be at x = 0. but that would not be an absolute or local max or min. so all you have left to do is check the endpoints. the one that gives the larger value is the absolute max and the one that gives the smaller value is the absolute min
• Jan 11th 2008, 05:06 AM
colby2152
Quote:

Originally Posted by Truthbetold
How do you do it?
Say the problem $\displaystyle \frac{1}[x^2}$.
The critical points are only a few.
How do you figure out whether this is the absolute or local without looking at the graph?

Thank you.

In case anyone was wondering, the LaTeX was supposed to be this: $\displaystyle \frac{1}{x^2}$

Take the derivative, and set it equal to zero to find ANY critical points (defined where the slope is zero)...

$\displaystyle \frac{-2}{x^3} = 0$

As you can see, no values of $\displaystyle x$ will make this derivative zero.