Differential Equations · Question 2

Let y=f(x) be the real valued function defined on the interval (0, ), satisfying y(1)=0 and the differential equation x dydx=y-x3. Then which of the following statements is (are) TRUE ?

1. The function f has a local minimum at x= 1 3
2. The function f has a local maximum at x= 1 3
3. The function f is increasing in the interval (1, 2)
4. If g(x)=4x3-5x2+ 32x for x>0, then the number of elements in the set \x (0, ): f(x)=g(x)\ is 2

Answer

(2) The function f has a local maximum at x= 1 3, (4) If g(x)=4x3-5x2+ 32x for x>0, then the number of elements in the set \x (0, ): f(x)=g(x)\ is 2

Solution

x y'-yx2=-x ddx ( yx )=-x yx=- x22+C; y(1)=0 C= 12.

f(x)= x-x32, f'(x)= 1-3x22, f''(x)=-3x.

f'=0 at x= 1 3 with f''<0: local maximum. (B) true, (A) false.

On (1,2), f'<0: decreasing. (C) false.

f=g x-x32=4x3-5x2+ 32x x(9x2-10x+2)=0; for x>0: x= 5 79, both positive, so 2 elements. (D) true.

Answer: (B), (D)