SOLUTION: Given that f"(x)={{{3x^2-2}}} , {{{f(2)=-1}}} , f'(-1)=5 , find {{{f(x)}}}. I need {{{steps}}}. {{{THANKS!}}}

Algebra ->  Expressions-with-variables -> SOLUTION: Given that f"(x)={{{3x^2-2}}} , {{{f(2)=-1}}} , f'(-1)=5 , find {{{f(x)}}}. I need {{{steps}}}. {{{THANKS!}}}      Log On


   

Question 702267: Given that f"(x)=3x%5E2-2 , f%282%29=-1 , f'(-1)=5 , find f%28x%29.
I need steps. THANKS%21
Answer by jim_thompson5910(35256) About Me  (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 f%28x%29+=+expr%281%2F4%29%2Ax%5E4+-+x%5E2+%2B+4x+-+9