cat 2021 Question 1

Question

f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if

Accepted Answer

$\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ < 0

$\frac{(x+5)(x-3)}{(x-9)(x+2)}$ < 0

We have four inflection points -5, -2, 3, and 9.

For x<-5, all four terms (x+5), (x-3), (x-9), (x+2) will be negative. Hence, the overall expression will be positive. Similarly, when x>9, all four terms will be positive.

When x belongs to (-2,3), two terms are negative and two are positive. Hence, the overall expression is positive again.

We are left with the range (-5,-2) and (3,9) where the expression will be negative.

Question 1.

Question Explanation