Critical Points

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One important application of calculus to science and engineering is finding the largest or smallest values a function achieves. Special points called critical points are useful here. A critical point of a function f is defined to be a point in the domain of f where either

  1. f' is zero, or
  2. f' doesn’t exist.

Let’s look at the first kind of critical point.

Consider the parabola f(x)=x^2. The derivative f'(x)=2x is zero when x=0, so this is a critical point of f. What happens at f near zero? Well, the derivative goes from being negative when x<0 to positive when x>0. This means that tangent lines to f point downward when x<0 and upward when x>0. It’s clear that there must be a minimum at x=0, and if you recall what the graph of the parabola x^2 looks like, indeed there is.

This is a simple example, but we can easily compute the critical points of more complicated functions. For example, the derivative of any polynomial has domain \mathbb{R} (all real numbers), so to compute the critical numbers we only have to find the zeros of the derivative.

What are the critical numbers of g(x)=e^x? The derivative is g'(x)=e^x. This function is defined for all real numbers, and since e^x\neq 0 for any value of x, g has no critical numbers.

What about h(x)=\cos x? The derivative is h'(x)=-sin x, and sin x=0 when x is an integer multiple of \pi. So h has infinitely many critical points, one at each integer multiple of \pi.