A curve with equation y=f(x) passes through point P at (4,8). Given that f'(x)=9x^(1/2)/4+5/2x^(1/2)-4 find f(X).

To find f(x) using its derivative, first integrate f'(x) with respect to x using the 'add one to the power and divide by the new power' technique remembering to add the constant c. To make this easier, turn the denominator x to a numerator with a negative power and let the square roots be shown as powers of a half. f(x) = integral f'(x) dx = (9/4)(2/3)x^(3/2)+(5/2)(2/1)x^(1/2)-4x+c = 3x^(3/2)/2+5x^(1/2)-4x+c Then find a value for c. To do this, substitute into f(x) the known value of x from the given point P and set it equal to the known value of y, also from point P. Then rearrange to solve for c. when x = 4, y = 8 f(x) = 12+10-16+c = 8c = 2Sub this found value back in to find f(x). f(x) = 3x^(3/2)/2 + 5x^(1/2) - 4x + 2