First slide

Mean value theorems

Question

f (x) is continuous on [0, 2] , differentiable on (0,2) , f(0) = 2 , f(2) = 8 and $f' (x) \leq 3$ for all x in (0,2) , then the value of f(1)

Moderate

A

2
B

3
C

4
D

5

Solution

$\begin{array}{l} f (0) = 2 f (2) = 8 f^{'} (x) \leq 3 \\ C_{1} \in (0, 1), f^{'} (x) = \frac{f (1) - f (0)}{1 - 0} \\ f^{'} (C_{1}) \leq 3 \Rightarrow \frac{f (1) - 2}{1} \leq 3 \\ f (1) \leq 5 \\ C_{2} \in (1, 2), f^{'} (x) = \frac{f (2) - f (1)}{2 - 1} \leq 3 \\ \frac{8 - f (1)}{1} \leq 3 \\ f (1) \geq 5 \\ \Rightarrow f (1) = 5 \end{array}$

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