Question

Calculus Help??????

Sketch the graph of a function f having the given characteristics.

f(0)=f(2)=0
f'(x)>0 if x<1
f'(1)=0
f'(x)<0 if x>1
f''(x)<0

I don't know how to do this. Please help??

Answer 1

A function similar to your function can be defined
\[
f(x) = 2 x - x^2\\
\]
You cam verify that it would satisfy all your conditions.
Graph it and you will be done.

Answer 2

Thank you!!!

Answer 3

Could you tell me how you did it? please?

Answer 4

@eliassaab

Answer 5

Let us try to see if we can find a parabola that satisfies your conditions:
\[
f(x) = a x^2 + b x + c \\
f(0)=0=c\\
c=0\\
f(x) = a x^2 + bx\\
f'(x) = 2 a x + b\\
f'(1)= 2 a + b =0\\
b=-2a\\
f(x)= a x^2 -2 a x\\
f''(x) = 2a < 0\\

\]
Take a = -1, or -2 or anything negative will do.
So f(x) can be taken
\[
f(x)= -x^2 + 2x
\]

Answer 6

So is it the first equation you gave me or the second?

Answer 7

Or is it the same equation, just rearranged?