The total number of local maxima and local minima of the function is
- A.0
- B.1
- C.2
- D.3
Show answer and solution
Correct option: C
Solution
At x = -1: from the left, (2+x)³ approaches (1)³ = 1, and from the right, x^(2/3) approaches 1, so f is continuous. For x slightly less than -1, (2+x)³ is increasing (derivative 3(2+x)² > 0), and for x slightly greater than -1, x^(2/3) is decreasing (derivative (2/3)x^(-1/3) < 0 for x < 0). So f increases then decreases → local maximum at x = -1.
At x = 0: the derivative of x^(2/3) is (2/3)x^(-1/3), which is negative for x < 0 and positive for x > 0 (within (-1,2)), so f decreases then increases → local minimum at x = 0.
These are the only two critical points with sign changes, giving a total of 2 local extrema.
At x = 0: the derivative of x^(2/3) is (2/3)x^(-1/3), which is negative for x < 0 and positive for x > 0 (within (-1,2)), so f decreases then increases → local minimum at x = 0.
These are the only two critical points with sign changes, giving a total of 2 local extrema.
[AIEEE 2008]