Answer the following question Show all work to receive full

Answer the following question. Show all work to receive full credit. Suppose that f is continuous on [0,4], f(0) = 1, and for all x in (0,4). Show that

Solution

From the fundamental theorem of calculus:

int f\'(x) dx (0 < x < 4) = f(x) (0 < x < 4) = f(4) - f(0)

We have:

f\'(x) <= 5 -> int f\'(x) dx <= int 5 dx (x from 0 to 4) = 5x (x from 0 to 4) = 5*4 - 5*0 = 20

So:

f(4) - f(0) <= 20

f(4) - 1 <= 20  

f(4) <= 21

On the other hand:

f\'(x) >= 2 -> int f\'(x) dx >= int 2 dx (x from 0 to 4) = 2x (x from 0 to 4) = 2*4 - 5*0 = 8

So:

f(4) - f(0) >= 8

f(4) - 1 >= 8

f(4) >= 9