Many first-year calculus students fall into a common trap: they tend to make bad assumptions about how functions behave. In particular, they tend to think all functions are "nice," in the sense of easy to draw and understand -- because most of the pictures their teachers draw in school to illustrate examples tend to be of nice, familiar functions they are comfortable working with, like polynomials. But functions, in general, are extremely unwieldy, and to truly master differential calculus, you have to learn to be on guard against making simplifying assumptions: what we often imagine to be the case turns out to be false on closer inspection.
Here's a simple example in the context of critical points and local extrema. A classic application of derivatives involves finding the local minima and maxima of a function. You may recall that the first step to finding these values is to find the function's critical points. Here's the common trap: most students mistakenly confuse "critical points of a function" with "points where the derivative of the function is 0." But an arbitrary function -- even an arbitrary continuous function -- need not be differentiable at every point in its domain! Critical points are NOT ONLY points where the derivative of a function is zero; they ALSO include points where the derivative is undefined. If you were looking for the local extrema of the absolute value function, for example -- the function f(x)=|x| -- you would not find its local (in fact, global) minimum at the origin if you looked only for points where f'(x) is zero, because there is no such point!
Who cares? Well, it's important to get this fact straight in any situation where you might be tested on the relationship between critical points and local extrema. To see if you're up to speed, check out the following worksheet I created below. If you work with me, you'll learn to solve all these problems and have access to many more specialized materials I've developed over the years, including problem sets and suggestions you won't find anywhere else.
PROBLEM SET: CRITICAL POINTS AND LOCAL EXTREMA
Problem 1. For each of the following claims, first decide whether it is true or false. If a claim is false, give a counterexample; if a claim is true, explain why.
a) If a function f(x) has a local extremum at x = c, then c is a critical point of f.
b) If a function f(x) has a local extremum at x = c, then f is differentiable at c.
c) If a function f(x) has a local extremum at x = c, then f is continuous at c.
d) If f'(c) = 0, then c is a critical point of f.
e) If f'(c) = 0, then f has a local extremum at x=c.
f) If f''(c) = 0, then c is a point of inflection of f.
g) The critical points of a function f(x) are those x values for which f'(x) = 0.
h) If f'(c)=0, then f(c) is either a local maximum or a local minimum.
i) If f''(c) = 0, then f'(c) is not zero.
j) If f''(c) = 0, then f'(c) = 0.
Problem 2. Fill in the blanks to make each sentence true.
a) In order for f to have a local extremum at c, c must be a ___________, but this requirement is only necessary and NOT sufficient.
b) Similarly, in order for f to have a point of inflection at c, c must satisfy ___________, but again this requirement is only necessary and NOT sufficient.
CHALLENGE PROBLEM, FOLLOW-UP: Can you find an example of a differentiable function that has a point of inflection (i.e., changes concavity) at the origin but whose second derivative is not defined there? (Hint: One way you might start is to think about what you might want the FIRST derivative of such a function to be.)
Problem 3. Which is the stronger condition: being a critical point, or being a local extremum? Which is the weaker condition: being a point of inflection, or being a point that satisfies f ''(c) = 0?