Question
Let R denote the set of all real numbers. Define the function f: R R by f( x )= \ array cc 2-2 x^2-x^2 1 x & if x 0 \\ 2 & if x=0 array . Then which one of the following statements is TRUE ?
JEE Advanced
Mathematics
Application of Derivatives
2025
JEE Advanced 2025 (Paper 1)
JEE Advanced Mathematics Question (2025) — Solution
Options
- A. The function f is NOT differentiable at x=0
- B. There is a positive real number , such that f is a decreasing function on the interval (0, )
- C. For any positive real number , the function f is NOT an increasing function on the interval (- , 0)
- D. x=0 is a point of local minima of f
Answer
C. For any positive real number , the function f is NOT an increasing function on the interval (- , 0)
Step-by-step solution
(A) RHD at x=0: _ h 0 (2-2 h^2-h^2 1 h )-2 h =0 Similarly LHD at x =0 is also equal to 0 . Differentiable at x =0 (B) f^ (x)=-4 x-2 x 1 x -x^2 ( 1 x ) ( -1 x^2 ) aligned & f^ (x)=- (4 x+2 x 1 x )+ 1 x \\ & f^ (x)=- (2 x (4- 1 x ) )+ 1 x aligned for x (0, ) We can't say f(x) is decreasing on (0, ) as 1 x oscillates. (C) for x (- , 0), for any >0 f(x) is not increasing on (- , 0) as 1 x oscillates from -1 to 1. (D) f(0)=2 aligned & f(0+h) <2 \\ & f(0-h) <2 aligned x =0 is local maxima
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