Optimization Problems

May 2010
I need an example or more of optimation problems - finding maximum values.

May 2010
wow, choose your own question :D

I'll make an easy one then.

Find the maximum of \(\displaystyle y=-(x-1)^2 + 15\)

Firstly, this is a familiar graph (upside down parabola, shifted around). The maximum value of y will be 15, occuring at x=1.

Now, to prove it:

Step 1: Find the derivative of the function you want to maximuse
\(\displaystyle f'(x)=-2(x-1) +0\)

Step 2: Set the derivative to zero to find x:
\(\displaystyle 0=-2(x-1)\)
\(\displaystyle x=1\)

Step 3: Find the value of y that corresponds to this value of x
\(\displaystyle y=-(x-1)^2 + 15\)
\(\displaystyle y=-0^2 + 15\)
\(\displaystyle y= 15\)

Step 4:Check you have a maximum
You can do this by checking the y values on either side are less than 15, or by checking that the second derivative is negative

We found the maximum by looking for the point where the gradient was 0. This is the point where the curve stops increasing/decreasing, so it will be a maximum or a minimum.