Trigonometric Functions · Question 24

The number of solutions of the equation 2x + 3 x = , x [-2 , 2 ] - \ 2, 3 2 \ is:

1. 4
2. 5
3. 3
4. 6

Answer

(2) 5

Solution

x = - 2x3: count the intersections of y = x with the decreasing line y = - 2x3.

On each branch x increases from - to while the line decreases ⇒ exactly one intersection on each complete branch.

Complete branches in the interval: (- 3 2, - 2 ), (- 2, 2 ), ( 2, 3 2 ) ⇒ 3 solutions.

On [-2 , - 3 2 ): at x = -2 , x = 0 < 5 3 (line); as x - 3 2-, x > 4 3 ⇒ 1 solution.

On ( 3 2, 2 ]: as x 3 2+, x - ; at x = 2 , x = 0 > - (line) ⇒ 1 solution.

∴ Total number of solutions = 3 + 1 + 1 = 5

Answer: (2)