Soon after being introduced to calculus, most students learn of its application to *optimization:* that is, to the problem of finding the largest or smallest value of a given differentiable function, which is usually referred to as the *objective function.* A very helpful observation is that if *f* is an objective function that is maximized or minimized at *x,* then the tangent to the graph at the point (*x*, *f*(*x*)) will be horizontal, since otherwise we can find some value *x*′ close to *x* for which *f*(*x*′) is higher. This means that we can narrow down the search for the maximum and minimum values of *f* by looking just at the values of *f*(*x*) for which *f*′(*x*) = 0.

Now suppose that we have ...

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