# 4.5: Optimization

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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).

## Optimization

The height of a baseball follows the function $$h(t) = -5t^2 + 20t + 10$$, where $$h$$ is measured in meters and $$t$$ is measured in seconds.. What value of $$t$$ maximizes the height?

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:

## Optimization 2

The cost per item of producing Super Hero Action Figures, if $$x$$ are produced, is given by

$$C(x) = \frac{500}{x} + 0.001x.$$

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!

This page titled 4.5: Optimization is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.