JEE Mains · Maths · STD 12 - 6. Application of derivatives
The function \(f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}\)
- A decreases in \((-2,8)\) and increases in \((-\infty,-2) \cup(8, \infty)\)
- B decreases in \((-\infty,-2) \cup(-2,8) \cup(8, \infty)\)
- C decreases in \((-\infty,-2)\) and increases in \((8, \infty)\)
- D increases in \((-\infty,-2) \cup(-2,8) \cup(8, \infty)\)
Answer & Solution
Correct Answer
(B) decreases in \((-\infty,-2) \cup(-2,8) \cup(8, \infty)\)
Step-by-step Solution
Detailed explanation
\(f(x)=\frac{x}{x^2-6 x-16}\) Now, \( \mathrm{f}^{\prime}(\mathrm{x})=\frac{-\left(\mathrm{x}^2+16\right)}{\left(\mathrm{x}^2-6 \mathrm{x}-16\right)^2} \) \( \mathrm{f}^{\prime}(\mathrm{x})<0\) Thus \(\mathrm{f}(\mathrm{x})\) is decreasing in…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Lets \(S=\{z \in C:|z-1|=1\) and \((\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2}\}\). Let \(\mathrm{z}_1, \mathrm{z}_2\) \(\in S\) be such that \(\left|z_1\right|=\max _{z \in S}|z|\) and \(\left|z_2\right|=\min _{z \in S}|z|\). Then \(\left|\sqrt{2} z_1-z_2\right|^2\) equals :JEE Mains 2024 Hard
- Let \(\mathrm{C}\) be the centroid of the triangle with vertices \((3,-1),(1,3)\) and \((2,4) .\) Let \(P\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 \mathrm{x}-\mathrm{y}+1=0 .\) Then the line passing through the points \(\mathrm{C}\) and \(\mathrm{P}\) also passes through the pointJEE Mains 2020 Hard
- If \(a, b\) and \(c\) are the greatest value of \(^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}\) and \(^{21 }\mathrm{C}_{\mathrm{r}}\) respectively, thenJEE Mains 2020 Hard
- The value of the integral \(\int \frac{\sin \theta \cdot \sin 2 \theta\left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} d \theta\) is (where \(c\) is a constant of integration)JEE Mains 2021 Hard
- Let \(\Omega\) be the sample space and \(A \subseteq \Omega\) be an event. Given below are two statements : \((S1)\) : If \(P ( A )=0\), then \(A =\phi\) \(( S 2)\) : If \(P ( A )=\), then \(A =\Omega\) ThenJEE Mains 2023 Hard
- If \(\overrightarrow{ a }=\alpha \hat{ i }+\beta \hat{ j }+3 \hat{ k }\) \(\overrightarrow{ b }=-\beta \hat{ i }-\alpha \hat{j}-\hat{ k }\) and \(\overrightarrow{ c }=\hat{ i }-2 \hat{ j }-\hat{ k }\) such that \(\overrightarrow{ a } \cdot \overrightarrow{ b }=1\) and \(\overrightarrow{ b } \cdot \overrightarrow{ c }=-3,\) then \(\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})\) is equal to ............JEE Mains 2021 Hard
More PYQs from JEE Mains
- If \(A\) is a symmetric matrix and \(B\) is a skew-symmetrix matrix such that \(A + B = \left[ {\begin{array}{*{20}{c}}
2&3\\
5&{ - 1}
\end{array}} \right]\) , then \(AB\) is equal toJEE Mains 2019 Hard - Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}x+a, & x<0 \\ |x-1|, & x \geq 0\end{array}\right.\) and \(g(x)=\left\{\begin{array}{cc}x+1, & x<0 \\ (x-1)^{2}+b, & x \geq 0\end{array}\right.\) where \(a , b\) are non-negative real numbers. If \((gof)\,( x )\) is continuous for all \(x \in R\), then \(a + b\) is equal to ...... .JEE Mains 2021 Hard
- For some \(\mathrm{a}, \mathrm{b}\), let \(f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim _{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}\). Then \((\lambda+\mu+v)^2\) is equal to :JEE Mains 2025 Medium
- The number of ordered pairs ( \(\mathrm{r}, \mathrm{k}\) ) for which \(6 \cdot ^{35} \mathrm{C}_{\mathrm{r}}=\left(\mathrm{k}^{2}-3\right)\cdot{^{36} \mathrm{C}_{\mathrm{r}+1}}\). where \(\mathrm{k}\) is an integer, isJEE Mains 2020 Hard
- Let \(H\) be the hyperbola, whose foci are \((1 \pm \sqrt{2}, 0)\) and eccentricity is \(\sqrt{2}\). Then the length of its latus rectum isJEE Mains 2023 Hard
- Let \(A =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) and \(B =\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]\) such that
\(AB = B\) and \(a + d =2021,\) then the value of \(ad - bc\) is equal to ...... .JEE Mains 2021 Medium