Question

Suppose that f(0)=1 and f′(x)≤3 for all values of x. Use the Mean Value Theorem to determine how large f(4) can possibly be

f(4)≤___________

Answer 1

@sourwing help

Answer 2

well, the mean value theorem says

f(b) - f(a) = f(c) (b-a)

where a = 0 and b = 4. What can you say?

Answer 3

?

Answer 4

f(4) - f(0) = f'(c) (4 - 0),

f(4) = f'(c) (4) + f(0)

you want to maximize f(4)

Answer 5

and the answer is?

Answer 6

you want f'(c) to be as big as possible right? so let f'(c) = 3

f(4) = 3(4) + 1 = 13

so f(4) <= 13

Answer 7

thank you

Answer 8

@sourwing, could you help me in a couple questions more?

Answer 9

no, but post your question and someone might answer it

Answer 10

thanks again

Answer 11

here is another way to think about this easy
at noon you are at mile marker one
you can walk no more than 3 mph
how far can you be at 4 pm?

Answer 12

you cannot have walked more than \(3\times 4=12\) miles so you cannot be past mile marker \(13\)