For the function f x = 4 log e x - 1 - 2 x 2 + 4 x + 5 , x > 1 , which one of the following is NOT…

For the function fx=4logex-1-2x2+4x+5,x>1, which one of the following is NOT correct?
  1. fx is increasing in 1,2  and decreasing in 2,∞
  2. fx=-1 has exactly two solutions
  3. f'e-f"2<0
  4. fx=0 has a root in the interval e,e+1

Solution

Given fx=4logex-1-2x2+4x+5,x>1

f'x=4x-1-4x-1

For 1<x<2⇒f'x>0

For x>2⇒f'x<0  

Hence, fx has a maxima at x=2

fx=-1 has two solution as the curve of fx will cut it at exactly two points.

Now, fe>0 and fe+1<0

i.e. fe·fe+1<0  

f'e-f"2=4e-1-4e-1+8>0

Asked in: JEE Main 2022 (24 Jun Shift 1)

Practice more Applications of Derivatives questions on Aicharya