Find f. f''(x) = 6 + 6x + (24x^2) , f(0) = 5, f(1) = 14
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You need to use the inverse of differentiation, hence you need to integrate the given function twice to find the original function f(x).
`int f''(x) dx = f'(x) + c`
`int (6 + 6x + 24x^2)dx = int 6dx + int 6x dx + int 24x^2 dx`
`int (6 + 6x + 24x^2)dx = 6x + 6x^2/2 + 24x^3/3 + c`
`int (6 + 6x + 24x^2)dx = 6x + 3x^2 + 8x^3 + c`
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integrate the f''(x) to get f'(x)
f''(x)=6+6x+(24x^2)
f'(x)=6x+3x^2 + 8x^3 + C
and i think you made a mistake because they should have given you a solution for the f'(x). Maybe you meant to write that f'(0)=5 and f(1)=14?
in that case...
f'(0)=6(0)+3(0^2)+8(0^3)+C=5
C=5
then:
f'(x)=6x+3x^2 + 8x^3 + 5
f(x)=3x^2 + x^3 + 2x^4 + 5x + C
f(1)=3(1)^2 + (1)^3 + 2(1)^4 + 5(1) + C=14
C=3
f(x)=3x^2 + x^3 + 2x^4 + 5x + 3
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