1. ## Problem involving f'

If f(4) = 5 and f '(x) ≥ 1 for 4x6, how small can f(6) possibly be?

Not sure where to start.

2. Draw a picture on the interval in question. Think about what the gradient of the function is doing over that interval

The gradient is positive increasing through the interval so min of f(6) has to also be > 5.

3. Originally Posted by hazecraze
If f(4) = 5 and f '(x) ≥ 1 for 4x6, how small can f(6) possibly be?

Not sure where to start.
Given that $f'(x)\geq 1$.

Now by Lagrange's Mean Value Theorem on the interval $[4,6],$we have

$f'(c)=\frac{f(6)-f(4)}{6-4}=\frac{f(6)-5}{2}\geq 1$,for some $c\in(4,6)$

$f(6)\geq 7$

Thus smallest possible value of $f(6)$ is $7$.