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Handout Wednesday 10/28
We will start exploring the big application of derivatives: maximizing and minimizing a function. This starts with section 3.3, and will continue with 3.5 and 3.6 (we will skip 3.4).
Today we start by looking for local extrema. What we explore today will not really feel like an application, but hopefully it is not too much of a leap to think that the functions we have here could describe some real-world quantity (say cost or revenue) that we would like to minimize or maximize.
Start by looking at a graph that has local maxima, minimal, cusp points, and a saddle point. Consider what each of these are in terms of local extrema, global extrema, and values of the derivative.
What can we learn from this example? What is the connection between the local extrema a critical numbers: inputs at which the function has either derivative zero or no derivative?
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If a function as a local max/min at input \(x = c\text{,}\) then either \(f'(c) = 0\) or \(f'(c)\) does not exist. We call such \(c\) a critical number, the point \((c, f(c))\) a critical point and the output \(f(c)\) a critical value
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However, just because \(c\) is a critical number, doesn’t mean that \(f\) has a local extrema at \(c\text{.}\)
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Why not? What can stop this from happening? This is where we use the first derivative near \(x = c\) to understand the behavior of the function.
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The first derivative test says that if \(p\) is a critical number of a continuous function \(f\) that is differentiable near \(p\) (except possibly at \(x=p\)), then we can look at the sign of \(f'\) on either side of \(p\) to determine whether \(f\) has a local maximum or minimum at \(p\text{.}\) In particular, if \(f'\) changes from positive to negative at \(p\text{,}\) then \(f\) has a local maximum at \(p\text{.}\) If \(f'\) changes from negative to positive at \(p\text{,}\) then \(f\) has a local minimum at \(p\text{.}\)
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We could also/alternatively use the second derivative test: if \(p\) is a critical number of a continuous function \(f\) such that \(f'(p) = 0\) and \(f''(p) \ne 0\text{,}\) then \(f\) has local maximum at \(p\) if and only if \(f''(p) \lt 0\) and \(f\) has a local minimum at \(p\) if and only if \(f''(p) \gt 0\text{.}\)
Now lets see how this all works with an example. Consider \(f(x) = 2x^3 + 3x^2 - 12x - 7\text{.}\) Find the critical numbers (factor), classify each using both tests, find points of inflection. Graph the function (by hand). Check that this makes sense by graphing with desmos.
Another example (needs quadratic formula) \(f(x) = x^3 + x^2 - 3x + 1\text{.}\)
