1
Let f be a differentiable function on R such that f(2)=1, f'(2)=4. Let x 0(f(2+x))3/x=e . Then the number of times the curve y=4x3-4x2-4( -7)x- meets x-axis is :
- 3
- 1
- 2
- 0
Answer
(3) 2
Solution
f(2)=1, the limit is of the form 1
x 0(f(2+x))3/x=e x 0 3(f(2+x)-1)x=e3f'(2)=e12
=12
y=4x3-4x2-20x-12=4(x3-x2-5x-3)
x=-1: -1-1+5-3=0, so (x+1) is a factor
y=4(x+1)(x2-2x-3)=4(x+1)2(x-3)
the curve meets the x-axis at x=-1 (touches) and x=3, i.e. at 2 points
Answer: (3)
JEE Main 2025 (Online) 4th April Evening Shift