Application of Derivatives · Question 17

Let f : R R be a function defined by f(x) = ||x + 2| - 2|x| |. If m is the number of points of local minima and n is the number of points of local maxima of f, then m + n is

1. 3
2. 4
3. 2
4. 5

Answer

(1) 3

Solution

Let g(x) = |x + 2| - 2|x|.

For x -2: g(x) = -(x + 2) + 2x = x - 2

For -2 x 0: g(x) = (x + 2) + 2x = 3x + 2

For x 0: g(x) = (x + 2) - 2x = 2 - x

So f(x) = |g(x)|:

x -2: f(x) = 2 - x, decreasing to f(-2) = 4

-2 x 0: f(x) = |3x + 2|, decreasing to 0 at x = - 23, then increasing to f(0) = 2

x 0: f(x) = |2 - x|, decreasing to 0 at x = 2, then increasing

⇒ At x = -2, f keeps decreasing on both sides, so it is not an extremum.

⇒ Local minima at x = - 23 and x = 2: m = 2

⇒ Local maximum at x = 0: n = 1

∴ m + n = 3

Answer: (1)