First slide

Mean value theorems

Question

Let the function $f : [- 7, 0] \to R$ be continuous on $[- 7, 0]$ and differentiable on $(- 7, 0)$ . If $f (- 7) = - 3$ and $f^{'} (x) \leq 2$ for all such function f, $f (- 1) + f (0)$ is in the interval:

Easy

A

$[- 3, 11)$
B

$(- \infty, 20]$
C

$(- 6, 20)$
D

$(- \infty, 11]$

Solution

Lets use LMVT for $x \in [- 7, - 1]$
   $\frac{f (- 1) - f (- 7)}{(- 1 + 7)} = f^{'} (x) \leq 2$
$\Rightarrow \frac{f (- 1) + 3}{6} \leq 2 \Rightarrow f (- 1) \leq 9 \to (1)$
Also use LMVT for $x \in [- 7, 0]$
   $\frac{f (0) - f (- 7)}{(0 + 7)} = f^{'} (x) \leq 2$
$\Rightarrow \frac{f (0) + 3}{7} \leq 2 \Rightarrow f (0) \leq 11 \to (2) A d d i n g (1) a n d (2), w e g e t$
     $∴ f (0) + f (- 1) \leq 20$

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