Question

If F Is A Differentiable Function Such That F′(X)≥1, Find The Largest Value Of M So That F(4)≥M Whenever F(2)=7.

M = ___________________

Answer 1

Using the Mean Value Theorem and information given, we find that the largest value of M for which F(4) ≥ M and F(2) = 7 is M = 9.

Step-by-step explanation:

The student is asking to find the largest value of M for which the inequality F(4) ≥ M holds true, given that F'(X) ≥ 1 and F(2) = 7. According to the Mean Value Theorem, there exists at least one point c in the interval (2, 4) such that F'(c) = (F(4) - F(2))/(4 - 2). Since this derivative is at least 1, we can multiply both sides of the inequality by 2 (the width of the interval) to get F(4) - F(2) ≥ 2. Substituting the given value of F(2) which is 7, we get F(4) ≥ 7 + 2, therefore M = 9 is the largest value satisfying the condition.