Question
Let f : R → R be defined by f x = x 2 - 3 x - 6 x 2 + 2 x + 4 Then which of the following statements is(are) TRUE?
JEE Advanced
Mathematics
Application of Derivatives
2021
JEE Advanced 2021 (Paper 1)
JEE Advanced Mathematics Question (2021) — Solution
Options
- A. f is decreasing in the interval ( - 2 , - 1 )
- B. f is increasing in the interval ( 1 , 2 )
- C. f is onto
- D. Range of f is - 3 2 , 2
Answer
B. f is increasing in the interval ( 1 , 2 )
Step-by-step solution
f x = x 2 - 3 x - 6 x 2 + 2 x + 4 f ' x = x 2 + 2 x + 4 2 x - 3 - x 2 - 3 x - 6 2 x + 2 x 2 + 2 x + 4 2 = 5 x 2 + 20 x x 2 + 2 x + 4 2 = 5 x ( x + 4 ) x 2 + 2 x + 4 2 Hence, it is increasing in ( - ∞ , - 4 ] ∪ [ 0 , ∞ ) Now as x → - ∞ then f x = 1 ,   also, x → ∞ then f x = 1 f - 4 = 16 + 12 - 6 16 - 8 + 4 = 22 12 = 11 6 f 0 = - 3 2 Hence, range is - 3 2 ,   11 6 , and the function is into.
Related: Mathematics — Application of Derivatives · All PYQ Banks