Question:

The maximum value of xexxe^{ -x} is

Updated On: Apr 16, 2024
  • ee
  • 1e\frac {1}{e}
  • e-e
  • 1e-\frac {1}{e}
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The Correct Option is B

Solution and Explanation

Let y=xex     y=x e^{-x}\,\,\,\,\,
On differentiating w.r.t. ' xx ', we get
dydx=xex(1)+ex\frac{d y}{d x}=x e^{-x}(-1)+e^{-x}
dydx=ex(1x)\frac{d y}{d x}=e^{-x}(1-x)
For maximum or minimum,
dydx=0\frac{dy}{dx}=0
ex(1x)=0\Rightarrow e^{-x}(1-x)=0
1x=0\Rightarrow 1-x=0
ex0)\because e^{-x} \ne 0)
x=1\Rightarrow x=1
From E (ii), we get
d2ydx2=ex(1)+(1x)ex(1)\frac{d^{2} y}{d x^{2}}=e^{-x}(-1)+(1-x) e^{-x}(-1)
=exex+xex=-e^{-x}-e^{-x}+x e^{-x}
=2ex+xex=-2 e^{-x}+x e^{-x}
d2ydx2=ex(x2)\frac{d^{2} y}{d x^{2}}=e^{-x}(x-2)
(d2ydx2)at x=1=e1(12)=1e(1)\left(\frac{d^{2} y}{d x^{2}}\right)_{\text {at } x=1} =e^{-1}(1-2)=\frac{1}{e}(-1)
Negative value
\therefore At x=1,yx=1, \,y is maximum and maximum value =1e1=1 e^{-1}
=1e=\frac{1}{e}
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Concepts Used:

Maxima and Minima

What are Maxima and Minima of a Function?

The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.

There are two types of maxima and minima that exist in a function, such as:

  • Local Maxima and Minima
  • Absolute or Global Maxima and Minima