A curve y = f(x) defined for x > 0 has the property that the slope of the tangent at any point (x, y) on it is given by y x + 2x^3 (x) . If the curve passes through the point (1, - 2 9 ) , then the local maximum value of the function z(x) = x^3 - 9y(x) 2x for x > 0 is

Step-by-step solution

The given geometric property can be written as the differential equation: dy dx = y x + 2x^3 (x) Rearranging into the standard linear form: dy dx - 1 x y = 2x^3 (x) The integrating factor (I.F.) is: I.F. = e^ - 1 x dx = e^ - (x) = 1 x Multiplying the differential equation by the I.F. and integrating, we get the general solution: y 1 x = (2x^3 (x) ) ( 1 x ) dx + C y x = 2x^2 (x) dx + C Using integration by parts (taking (x) as the first function and 2x^2 as the second): y x = (x) ( 2x^3 3 ) - 1 x ( 2x^3 3 ) dx + C y