JEE Main 2022 · 24 Jun · Shift 1 · Mathematics Q10

← JEE Main 2022 · 24 Jun · Shift 1

MathematicsSingle correct+4 / -1Question 10 of 90

For the function
$f(x)=4\log_e(x-1)-2x^2+4x+5,\quad x>1,$
which one of the following is NOT correct ?

f

is increasing in

(1,2)

and decreasing in

(2,\infty)

f(x)=-1

has exactly two solutions

f'(e)-f''(2)<0

f(x)=0

has a root in the interval

(e,e+1)

Answer and worked solution

Official answer: C

Derivative. $f'(x)=\dfrac4{x-1}-4x+4=\dfrac{4\left[1-(x-1)^2\right]}{x-1}$ , positive on $(1,2)$ , negative on $(2,\infty)$ . So (A) is true.

(B). The maximum is $f(2)=5$ and $f\to-\infty$ at both ends, so $f=-1$ has exactly two solutions: true.

(C). $f''(x)=-\dfrac4{(x-1)^2}-4$ , so $f''(2)=-8$ . Also $f'(e)=\dfrac4{e-1}-4e+4\approx-4.55$ . Then $f'(e)-f''(2)\approx3.45>0$ , so (C) is false.

(D). $f(e)\approx3.26>0$ and $f(e+1)\approx-3.78<0$ , so there is a root: true.

(C)

Topics:Monotonicity Maxima and minima

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JEE Main 2022 · 24 Jun · Shift 1, Mathematics question 10