Answer:
f(x) = x^4 +x^3 -2x^2 -8x +9
Explanation:
You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...
f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant
The second antiderivative is ...
f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant
Then the constants can be found from ...
f(0) = q = 9
f(1) = 1 + 1 - 2 +p + 9 = 1
p = -8
The solution is ...
f(x) = x^4 +x^3 -2x^2 -8x +9
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The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.