Browse Questions

# If $f(1)=10$ and $f '(x)\geq 2$ for $1\leq x \leq 4$ how small can $f(4)$ possibly be?

Toolbox:
• Lagrange's Mean Value Theorem :
• Let $f(x)$ be a real valued function that satisfies the following conditions.
• (i) $f(x)$ is continuous on the closed interval $[a,b]$
• (ii) $f(x)$ is differentiable in the open interval $(a,b)$
• (iii) $f(a)=f(b)$
• Then there exists atleast one value $c \in (a,b)$ such that $f'(c)=0$
$f(1)=10, f'(x) \geq 2, 1 \leq x \leq 4$
Step 1:
Assuming the condition's for Lagrange's theorem are satisfied in $[1,4]$ there is a $c \in (1,4)$ Such that $f'(c)=\large\frac{f(4)-f(1)}{4-1}$
Step 2: