4.5: Optimization
Sometimes we have a function and we just really want to know what its high point or low point is in terms of \(y\)-value. The high point is called a maximum, and the low point is called a minimum. For most functions, these points occur when the derivative is zero or undefined (we talked about why this is briefly in a previous section).
We will follow the maxim “optimization happens when the derivative is zero”. First we find the derivative using the power rule \(h'(t) = -10t + 20\). Then we set this equal to zero, so we solve
\[\begin{align*} -10t + 20 & = 0 \\ -10t & = - 20 \\ t & = \boxed{2} \end{align*}\]
Hence, the height is maximized when the time is equal to \(2\) seconds. At this point, the height of the ball is \(h(2) = -5(2)^2 + 20(2) + 10 = -20 + 40 + 10 = 30\) meters. Here’s a rough sketch based on what we know about this function:
Let’s do anther optimization example:
\(C(x) = \frac{500}{x} + 0.001x.\)
At what value \(x\) is the cost per item minimized? What is the cost at this value of \(x\)?
To solve this problem, we find \(C'(x)\):
\(C'(x) = \frac{d}{dx} 500 x^{-1} + 0.001 x = -500x^{-2} + 0.001\)
We then set this equal to zero and solve:
\[\begin{align*} -500x^{-2} + 0.001 & = 0 \\ -500x^{-2} & = -0.001 \\ x^{-2} & = 0.000002 \\ 1 & = 0.000002 x^2 \\ 500,000 & = x^2 \\ \sqrt{500,000} & = x \approx \boxed{707.10} \end{align*}\]
So we see about \(707\) action figures is the best number to choose. The cost at this point (by plugging \(x = 707\) into the original equation) is just \(\boxed{\$1.41}\). Not too bad!