Question

if f(1)=3 and f'(x)>(equal) 2 for 1<(eq)x<(eq)5, how small can f(5) possibly be?

Answer 1

\[f(1)=3\]
start at (1,3)
you know the slope is at least 2 so just think of a line with m = 2. that is the smallest m can be.
equation for that line would be
\[y-3=2(x-1)\]
\[y=2(x-1)+3\] so smallest y can be if x = 5 (i.e. f(5)) is
\[2(5-1)+3=8+3=11\]

Answer 2

or think this. from 1 to 5 is 4 units. the slope is at least 2 so you have gone at least 8 units. you start at 3, you are now at least 11