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MathCalculus

Mean Value Theorem (MVT): Find c for f(x)=x³ on [0,2]

Check the Mean Value Theorem on [0,2] for x³, then find every interior point whose tangent has the secant slope.

Question

For f(x)=x3f(x)=x^3, find every cc with 0<c<20<c<2 such that f(2)−f(0)=f′(c)(2−0)f(2)-f(0)=f'(c)(2-0), as predicted by the Mean Value Theorem.

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Evidence boundary

The exercise and interval come from OpenStax Exercise 161. The theorem verifies existence; the displayed algebra finds the only admissible point. No graph reading or measurements are needed.