Q. 10 - Calculus

Question

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

Step-by-Step Solution

Answer

The restated Fermat's theorem in terms of tangent line is ;

If x = c is the location of a local extremum of f , then x = c is a critical point of f .

Step 1. Given information .

Actual definition of Fermat's theorem .

If x = c is the location of a local extremum of f , then x = c is a critical point of f .

Step 2. Rewrite the Fermat's theorem .

The converse of this theorem is not true. That is, not every critical point is a local extremum of f . If f is continuous on $\left[a,b\right]$ then f attains its extreme value . If f has a local maximum and minimum at c and if f'(c) exist then f'(c) is equal to zero .