Apentix/AP Calculus AB/Unit 5: Analytical Applications of Differentiation/Question
AP Calculus ABhardmcq1 pt

Let f(x)=x2−4x+7f(x)=x^2-4x+7 on [1,4][1,4]. Which statement correctly applies the Mean Value Theorem?

A.ff is continuous on [1,4][1,4] and differentiable on (1,4)(1,4), so there is a cc in (1,4)(1,4) with f′(c)=f(4)−f(1)3=1f'(c)=\dfrac{f(4)-f(1)}{3}=1; then 2c−4=12c-4=1 gives c=52c=\dfrac{5}{2}.
B.There is a cc in (1,4)(1,4) with f′(c)=f(4)−f(1)3=3f'(c)=\dfrac{f(4)-f(1)}{3}=3, so c=72c=\dfrac{7}{2}.
C.Since f(1)≠f(4)f(1)\neq f(4), the Mean Value Theorem does not apply to ff on [1,4][1,4].
D.There is a cc in (1,4)(1,4) with f′(c)=1f'(c)=1; then 2c−4=12c-4=1 gives c=2c=2.

Explanation

Core Concept

A polynomial satisfies both MVT hypotheses on any interval. Computing f(1)=4f(1)=4 and f(4)=7f(4)=7 gives the average rate (7−4)/3=1(7-4)/3=1, and 2c−4=12c-4=1 gives c=5/2c=5/2.

Correct Answer

Aff is continuous on [1,4][1,4] and differentiable on (1,4)(1,4), so there is a cc in (1,4)(1,4) with f′(c)=f(4)−f(1)3=1f'(c)=\dfrac{f(4)-f(1)}{3}=1; then 2c−4=12c-4=1 gives c=52c=\dfrac{5}{2}.

More Unit 5: Analytical Applications of Differentiation practice questions

Try a random question →

Practice more AP Calculus AB questions with full explanations

Practice Unit 5: Analytical Applications of Differentiation Questions →