Application of Derivatives · Question 4

Let R denote the set of all real numbers. Let f: R R be defined by f(x)= cases 6x+ x2x+ x & if x 0,\\[2mm] 73 & if x=0. cases Then which of the following statements is (are) TRUE?

1. The point x=0 is a point of local maxima of f
2. The point x=0 is a point of local minima of f
3. Number of points of local maxima of f in the interval [ ,6 ] is 3
4. Number of points of local minima of f in the interval [2 ,4 ] is 1

Answer

(2) The point x=0 is a point of local minima of f, (3) Number of points of local maxima of f in the interval [ ,6 ] is 3, (4) Number of points of local minima of f in the interval [2 ,4 ] is 1

Solution

f'(x)= (6+ x)(2x+ x)-(6x+ x)(2+ x)(2x+ x)2= 4( x-x x)(2x+ x)2; f is continuous at 0 ( = 73).

Near 0: x-x x x33, so f'<0 for x<0, f'>0 for x>0: x=0 is a local minimum.

For x>0, critical points: x=x, one root in each (n ,n + 2 ): x 4.49, 7.73, 10.90, 14.07, 17.22.

h(x)= x-x x has h(n )=(-1)n+1n , so f' changes + - at 4.49, 10.90, 17.22 (maxima) and - + at 7.73, 14.07 (minima).

In [ ,6 ] (6 18.85): 3 maxima. In [2 ,4 ] (4 12.57): 1 minimum (7.73).

Answer: (B), (C), (D)