If f(4) = 5 and f '(x) ≥ 1 for 4 ≤ x ≤ 6, how small can f(6) possibly be?
Not sure where to start.
Given that $\displaystyle f'(x)\geq 1$.
Now by Lagrange's Mean Value Theorem on the interval $\displaystyle [4,6],$we have
$\displaystyle f'(c)=\frac{f(6)-f(4)}{6-4}=\frac{f(6)-5}{2}\geq 1$,for some $\displaystyle c\in(4,6)$
$\displaystyle f(6)\geq 7$
Thus smallest possible value of $\displaystyle f(6)$ is $\displaystyle 7$.