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Functions, derivatives, and critical numbers

Please show me how to reach correct solutions. Please see the attached file.
Thanks in advance,
Find f(x), given that...
Given that f is a differentiable function and the graph of its derivative is shown below, find the interval(s) in which f is decreasing.
Find the critical numbers...

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Please see the attached file for answers. Thanks.

1. Find f(x), given that

for all x and

Answer:

First integrate the expression using the power rule and trig identities. (hint: the derivative of the sin function is the cosine function)

f(x) = 10x2 + 3 sin(x)+ C

Now plug in f(0) = 2 and solve for C

f(0) = 10(0)2 + 3 sin(0)+ C = 2
0 + 0 + C = 2
C = 2

So the total equation now becomes

f(x) = 10x2 + 3 sin(x)+ 2

2. Given that f is a differentiable function and the graph of its derivative is shown below, find the interval(s) in which f is decreasing.

Answer:

We can look at the graph and see it is ...

Solution Summary

This shows how to find a function from the derivative and the graph of the derivative, as well as finding critical numbers and extrema.

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