wow, choose your own question

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.