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

Thanks

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- May 23rd 2010, 01:36 PMlorica2000Optimization Problems
I need an example or more of optimation problems - finding maximum values.

Thanks - May 23rd 2010, 03:37 PMSpringFan25
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

**Explanation**

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.