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Find f(x) given that f''(x) = 6 + 6x + 24x^2,  f(0) = 5 and f(1) = 14

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yakar | Student | eNoter

Posted April 13, 2012 at 5:41 PM via web

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Find f(x) given that f''(x) = 6 + 6x + 24x^2,  f(0) = 5 and f(1) = 14

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 13, 2012 at 5:52 PM (Answer #1)

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The second derivative of f(x) is f''(x) = 6 + 6x + 24x^2.

f'(x) = `int 6 + 6x + 24x^2 dx`

=> `6x + 3x^2 + 8x^3 + C1`

f(x) = `int 6x + 3x^2 + 8x^3 + C1 dx`

=> `3x^2 + x^3 + 2x^4 + C1*x + C2`

To determine C1 and C2 use the fact that f(0) = 5 and f(1) = 14

f(0) = C2 = 5

f(1) = 3 + 1 + 2 + C1 + 5 = 14

=> C1 = 3

The function f(x) = 3x^2 + x^3 + 2x^4 + 3x + 5

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