# Optimization

#### SyNtHeSiS

When doing an optimization problem, when would you use the closed interval method to get your values? Also if you obtain a critical point where the first derivative is 0, how would you know if this is a max or min?

#### HallsofIvy

MHF Helper
When doing an optimization problem, when would you use the closed interval method to get your values?
Only if the problem were given on a closed interval! Of course, if the interval is not closed and bounded, there might not be a "max" or "min".

Also if you obtain a critical point where the first derivative is 0, how would you know if this is a max or min?
You read in the next couple of pages of your text book about the first and second derivative tests!

first derivative test If $$\displaystyle x_0$$ is a critical point and f'(x)< 0 for $$\displaystyle x<x_0$$ and f'(x)> 0 for $$\displaystyle x> x_0$$, $$\displaystyle f(x_0)$$ is a minimum.
(Think about what f'> 0 and f'< 0 mean.)

For example, $$\displaystyle f(x)= x^2$$ has $$\displaystyle f'(x)= 2x$$ and so has x= 0 as a critical point. For x< 0, f'(x)= 2x< 0 while for x> 0, f'(x)= 2x> 0 so x= 0 gives f(0)= 0, a minimum.

If $$\displaystyle x_0$$ is a critical point and f'(x)> 0 for $$\displaystyle x<x_0$$ and f'(x)> 0 for $$\displaystyle x< x_0$$, $$\displaystyle f(x_0)$$ is a maximum.

For example, $$\displaystyle f(x)= -x^2$$ has $$\displaystyle f'(x)= -2x$$ and so has x= 0 as a critical point. For x< 0, f'(x)= -2x> 0 while for x> 0, f'(x)= -2x0 0 so x= 0 gives f(0)= 0, a maximum.

second derivative test If $$\displaystyle x_0$$ is a critical point and $$\displaystyle f''(x_0)> 0$$ then $$\displaystyle f(x_0)$$ is a minimum.

For example, for $$\displaystyle f(x)= x^2$$, f''(x)= 2> 0 so the critical point x= 0 gives a minimum value.

If $$\displaystyle x_0$$ is a critical point and f''< 0[/tex] then $$\displaystyle f(x_0)$$ is a maximum.

for example, for $$\displaystyle f(x)= -2x^2$$, f''(x)= -2< 0 so the crtical point x= 0 gives a maximum value.

Of course, it might happen that none of those conditions hold- that is that f' does not change sign or that f''= 0. In that case, it might be that the critical point is neither a maximum nor a minimum or it might be that you need some other test.

#### SyNtHeSiS

I know that you had to do that but like would you have to test values with a sign table to the left, between, and right of your critical points with the original function that you differentiated?

Also I know that if f''(x) is + then the graph is concave up and if f''(x) is - then concave down, but I dont understand this in terms of the 2nd derivative test graphically. Must you just memorize this rule and say if critical point + it min, and if - then max?