Question

Suppose the function f is continuous and differentiable for all real numbers. Prove that if f has 2 real roots, then f'(x) μst have at least one root specific term.

a) Critical point
b) Inflection point
c) Saddle point
d) Stationary point

Answer 1

Using the Intermediate Value Theorem and Rolle's Theorem, it can be shown that for a differentiable function with two real roots, there must be at least one stationary point between those roots where the first derivative f'(x) equals zero.

Step-by-step explanation:

The student is asking for a proof involving a continuous and differentiable function, specifically relating to the roots of the function and the roots of its derivative. If a continuous function f has two real roots, then by the Intermediate Value Theorem, f(x) attains every value between f(a) and f(b), where a and b are the roots.

Since f is differentiable (and thus continuous), Rolle's Theorem can be applied, stating that there exists at least one c in the interval (a, b) such that f'(c) = 0. This point, where the first derivative is zero, is known as a stationary point.

To prove this, let's assume f(a) = f(b) = 0, for a < b. Since f is continuous on [a, b] and differentiable on (a, b), Rolle's Theorem guarantees that there is at least one c in (a, b) such that f'(c) = 0.

This c corresponds to a stationary point because at c the function has a horizontal tangent, not increasing or decreasing.