Differentiation and its Application: The function f(x)=fracxx^2-6x-16 , xinmathbb R-\-2,8\ | JEE Main 2024

JEE Main 2024 (Online) 29th January Evening ShiftQuestion 1

The function $f(x)=\frac{x}{x^{2}-6x-16}$, $x\in\mathbb R-\{-2,8\}$

Differentiation and its Application

decreases in $(-2,8)$ and increases in $(-\infty,-2)\cup(8,\infty)$decreases in $(-\infty,-2)\cup(-2,8)\cup(8,\infty)$✓ Correct answerdecreases in $(-\infty,-2)$ and increases in $(8,\infty)$increases in $(-\infty,-2)\cup(-2,8)\cup(8,\infty)$

Solution

Concept. Compute $f'$; its numerator is always negative. Formulas used. (1) $\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}$Step 1 - $f'$. By (1), $$f'(x)=\frac{(x^2-6x-16)-x(2x-6)}{(x^2-6x-16)^2}=-\frac{x^2+16}{(x^2-6x-16)^2}.$$ Step 2 - Sign. $f'<0$ wherever it is defined, so $f$ decreases on each of $(-\infty,-2)$, $(-2,8)$, $(8,\infty)$. Answer: (B)

Differentiation and its Application - JEE Mathematics question (JEE Main 2024 (Online) 29th January Evening Shift)

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