Question 998629: Find the function f that satisfies the following conditions, f"(x)=6x+2, f(1)=2, f'(1)=-3
I'm having trouble with this one please help walk me through it.
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! Find the function f that satisfies the following conditions, f"(x)=6x+2, f(1)=2, f'(1)=-3
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Solve (d^2 f(x))/(dx^2)= 6 x+2 considering that f(1)= 2 and f'(1)= -3:
Let's integrate both sides with respect to x:
(df(x))/(dx)= integral (6x+2)dx= 3x^2+2x+c_1, where c_1 is an arbitrary constant.
Substitute f'(1)= -3:
c_1+5= -3
Solve for c_1:
c_1= -8
Substitute the value of c_1:
(df(x))/(dx)= 3 x^2+2 x-8
Integrate both sides with respect to x:
f(x)= integral (3x^2+2x-8)dx= x^3+x^2-8x+c_2, where c_2 is an arbitrary constant.
Substitute f(1)= 2:
c_2-6= 2
Solve for c_2:
c_2= 8
Substitute the value of c_2 and we get:
f(x)= x^3+x^2-8x+8
J
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