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