SOLUTION: Given that f"(x)={{{3x^2-2}}} , {{{f(2)=-1}}} , f'(-1)=5 , find {{{f(x)}}}.
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Question 702267: Given that f"(x)= , , f'(-1)=5 , find .
I need .
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
f '' (x) = 3x^2 - 2
Integrate both sides to get
f ' (x) = x^3 - 2x + C
We're told that f ' (-1) = 5, so
f ' (x) = x^3 - 2x + C
f ' (-1) = (-1)^3 - 2(-1) + C
5 = -1 + 2 + C
5 = 1 + C
5 - 1 = C
4 = C
C = 4
So f ' (x) = x^3 - 2x + 4
Integrate both sides to get
f(x) = (1/4)*x^4 - x^2 + 4x + C
Now use the fact that f(2) = -1
f(x) = (1/4)*x^4 - x^2 + 4x + C
f(2) = (1/4)*(2)^4 - (2)^2 + 4(2) + C
f(2) = (1/4)*(16) - 4 + 4(2) + C
f(2) = 4 - 4 + 8 + C
f(2) = 8 + C
-1 = 8 + C
-1 - 8 = C
-9 = C
C = -9
Therefore, the function f(x) is
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