AP EAMCET · Maths · Application of Derivatives
Observe the statements given below :
Assertion (A) : \(f(x)=x e^{-x}\) has the maximum at \(x=1\)
Reason (R) : \(f^{\prime}(1)=0\) and \(f^{\prime \prime}(1) < 0\)
Which of the following is correct?
- A Both \((A)\) and (R) are true and (R) is the correct reason for (A)
- B Both \((A)\) and \((R)\) are true, but (R) is not the correct reason for (A)
- C (A) is true, (R) is false
- D (A) is false, (R) is true
Answer & Solution
Correct Answer
(A) Both \((A)\) and (R) are true and (R) is the correct reason for (A)
Step-by-step Solution
Detailed explanation
Given, \(f(x)=x e^{-x}\) \(\begin{aligned} f^{\prime}(x) & =e^{-x}-x e^{-x} \\ f^{\prime \prime}(x) & =-e^{-x}-e^{-x}+x e^{-x} \\ & =-2 e^{-x}+x e^{-x}\end{aligned}\) For maximum, put \(f^{\prime}(1)=0 \Rightarrow x=1\) and \(f^{\prime \prime}(1)=-1 < 0\) \(\therefore\) Both…
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
- Two dice are rolled. Then, the probability that the total score is a prime number isAP EAMCET 2022 Easy
- Let \([x]\) denote the greatest integer not exceeding \(x\). If \(l_1=\lim _{x \rightarrow 2^{+}}\left(x^2+[x]\right)\),
\(l_2=\lim _{x \rightarrow 3^{-}}(2 x-[x])\) and \(l_3=\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\cos x}{x-\frac{\pi}{2}}\right)\), thenAP EAMCET 2019 Easy - General solution of \(4 \sin ^2(x)-4 \sin (x)+1=0\) isAP EAMCET 2020 Easy
- If a plane passes through \((1,-2,1)\) and is perpendicular to the planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of that plane from the point \((1,2,2)\) isAP EAMCET 2017 Medium
- If \(z_1, z_2\) are two complex numbers satisfying \(\left|\frac{z_1-3 z_2}{3-z_1 \bar{z}_2}\right|=1,\left|z_1\right| \neq 3\), then \(\left|z_2\right|\) is equal toAP EAMCET 2004 Medium
- The mean deviation from the median for the following data is
AP EAMCET 2022 Medium
More PYQs from AP EAMCET
- If \(y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)\), then \(\lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)=\)AP EAMCET 2024 Easy
- \(\int \frac{x+1}{x\left(1+x e^x\right)} d x\) is equal toAP EAMCET 2015 Medium
- Two copper wires A and B of lengths in the ratio \(1: 2\) and diameters in the ratio \(3: 2\) are stretched by forces in the ratio \(3: 1\). The ratio of the elastic potential energies stored per unit volume in the wires A and B isAP EAMCET 2024 Easy
- A Carnot engine takes \(3 \times 10^6\) calories of heat from reservoir at \(627^{\circ} \mathrm{C}\) and gives it to a sink at \(27^{\circ} \mathrm{C}\). The work done by the engine isAP EAMCET 2020 Medium
- The set \(\left\{x \in \mathbb{R}: 4+11 x-3 x^2>0\right\}\) is the intervalAP EAMCET 2023 Easy
- The rate constant is same for reactions of order , and , respectively, the unit of concentration being in moles per litre. If the concentration of the reactant is unity, the rates of reaction will beAP EAMCET 2020 Easy