Question

Let $f: R \rightarrow R$ 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?

• 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 $\left[-\frac{3}{2}, 2\right]$

Answer 1

Answer: (A), (B)

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

Answer 2

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

$⇒$ $f’(x) =$ $\frac{{5\times(x+4)}}{{(x^2 + 2x + 4)^2}}$

$⇒ f(x)$ has local maxima at x = -4 and minima at x = 0
Range of $f(x)$ is $\left[-\frac{3}{2}, \frac{11}{6}\right]$