Question

Suppose we know that $f\left(x\right)$ is continuous and differentiable on the interval $[-7,0]$, that $f(-7)=-3$ and that $f^{\prime }\left(x\right)\le 2$. What is the largest possible value for $f\left(0\right)$?

• A. $12$
• B. $22$
• C. $11$
• D. $24$

Answer 1

Answer: (C)

$11$

$f\left(x\right)$ is continuous and differentiable on the interval $[-7,0]

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$=\frac{f\left(0\right)-f(-7)}{7}$

$f^{\prime }\left(x\right)\le 2$

$\frac{f\left(0\right)-f(-7)}{7}\le 2$

$f\left(0\right)+3\le 14$

$f\left(0\right)\le 11$