A differentiable function f satisfies 1 f (x) 3 for all real numbers x . Moreover, it is known that f (2) = 4 . Use the Mean Value Theorem to answer…

A differentiable function f satisfies 1 ≤ f ′(x) ≤ 3 for all real numbers x .

Moreover, it is known that f (2) = 4 . Use the Mean Value Theorem to answer the following.

(a) What is the largest possible value of f(7) ? Explain.

(b) What is the smallest possible value of f(−4) ? Explain. 

I am not too sure how to approach this, originally what I did was use the Mean Value Theorem where I did f'(x) = f(2) – f(7) / 2 – 7 and then plug this into the inequality given in the question and then solved for the inequality including f(7) ? Can you please provide the solution and guidance? Thank You!

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