QJEEAdv21Ma4f2c: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma4f2c Solution: Let f : R R be defined by Then which of the following…

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be defined by

f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}

Then which of the following statements is (are) TRUE?

Accepted Answer

Given,

f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}

.... (i)

\Rightarrow f'(x) = {{({x^2} + 2x + 4)(2x - 3) - ({x^2} - 3x - 6)(2x + 2)} \over {{{({x^2} + 2x + 4)}^2}}}

\Rightarrow f'(x) = {{5x(x + 4)} \over {{{({x^2} + 2x + 4)}^2}}}

Sign scheme for f'(x)

Here, f is decreasing in the interval (

-

2,

-

1) and f is increasing in the interval (1, 2).

Now,

f( - 4) = {{11} \over 6},f(0) = {{ - 3} \over 2}

[from Eq. (i)]

and

\mathop {\lim }\limits_{x \to \pm \,\infty } f(x) = 1

\therefore

Range

= \left[ {{{ - 3} \over 2},{{11} \over 6}} \right]

Hence, f(x) is into.

JEE Advanced 2021 Paper 1 Online Mathematics - Limits, Continuity and Differentiability Question 26 English Explanation 2

f(x) has local maxima at x =

-

4

and local minima at x = 0.

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