Let f: R arrow R be defined by f(x)=(x2-3 x-6/x2+2 x+4). Then which of the following statements is(are) TRUE?

Question

Let f: R arrow R be defined by f(x)=(x2-3 x-6/x2+2 x+4). Then which of the following statements is(are) TRUE? (A) f is decreasing in the interval (-2,-1) (B) f is increasing in the interval (1,2) (C) f is onto (D) Range of f is [-(3/2), 2]

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/96693fab567a4229ecb4029d4057042a-.png" /> f prime(x)=(5 x(x+4)/(x2+2 x+4)2) f(0)=-(3/2) (point of local minima) f(-4)=(11/6) (point of local maxima)

Q. Let $f : R \to R$ be defined by $f (x) = \frac{x ^{2} - 3 x - 6}{x ^{2} + 2 x + 4}$ . Then which of the following statements is(are) TRUE?

$f$ is decreasing in the interval $(- 2, - 1)$

$f$ is increasing in the interval $(1, 2)$

$f$ is onto

Range of $f$ is $[- \frac{3}{2}, 2]$

$f^{'} (x) = \frac{5 x ( x + 4 )}{( x ^{2} + 2 x + 4 ) ^{2}}$
$f (0) = - \frac{3}{2}$ (point of local minima)
$f (- 4) = \frac{11}{6}$ (point of local maxima)

Q. Let f:R→R be defined by f(x)=x2+2x+4x2−3x−6​. Then which of the following statements is(are) TRUE?

f is decreasing in the interval (−2,−1)

f is increasing in the interval (1,2)

f is onto

Range of f is [−23​,2]