A curve with equation y = f(x) passes through the point (4,25). Given that f'(x) = (3/8)*x^2 - 10x^(-1/2) + 1, find f(x).
f'(x) = (3/8)x^2 - 10x^(-1/2) + 1Each term must be integrated (increase the power by 1 and divide by the new power), remembering to include + c.f(x) = (3/8)(x/3)^3 - 10*(2x)^(1/2) + x + cf(x) = (1/8)x^3 - 20 x^(1/2) + x + c = ySubstitute the given values for x and y into the equation, rearrange to find c.25 = (1/8)4^3 - 20 4^(1/2) + 4 + c c = 53Therefore f(x) = (1/8)*x^3 - 20x^(1/2) + x + 53
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