Method For Finding Maxima and Minima

When to use: f is continuous on its domain I, and f has exactly one interior critical point.

This method can be applied to an open or half-open interval as well as a closed interval. The Second Derivative Test is more convenient because it requires only the single computation f"(c), while the Direct Test requires the three computations f(u), f(v), and f(c). However, the Direct Test always works while the Second Derivative Test sometimes fails.

We illustrate the use of both tests in the examples.

Example 4

Example 5

Example 6

Example 7

The Critical Point Theorem can often be used to show that a curve has no maximum or minimum on an open interval I = (a, b). The theorem shows that:

If y = f(x) has no critical points in (a, b), the curve has no maximum or minimum on (a, b).

If y = f(x) has just one critical point x = c in (a, b) and two points x₁ and x₂ are found where f(x₁) < f(c) < f(x₂), then the curve has no maximum or minimum on (a, b).

Example 8