Question

For the function $f(x) = x{e^x}$ the point
A. $x = 0$ is a maximum
B. $x = 0$ is a minimum
C. $x = - 1$ is a maximum
D. $x = - 1$ is a minimum

Answer 1

Hint: Equating the first order derivative of the given function to zero to get the solution, and then substituting values of the solution in the second derivative of the function to discuss the point of maxima or minima. Finally substituting values in the given function to get maximum or minimum value corresponding to point of maxima or point of minima.

Formula Used:
$\dfrac{{df}}{{dx}} = 0$, $\dfrac{{{d^2}f}}{{d{x^2}}} < 0$ for point of maxima and $\dfrac{{{d^2}f}}{{d{x^2}}} > 0$ for point of minima.

Complete step by step Solution:
Maxima will be the highest point on the curve within the given range and minima would be the lowest point on the curve. Maxima and the minima of a function can be calculated by using the first-order derivative test and second-order derivative test. Derivative tests are the quickest ways to find the maxima and minima of a function There can only be one absolute maxima of a function and one absolute minimum of the function over the entire domain.

Given, function is $f(x) = x{e^x}$
Calculating first derivative of $f(x)$ and equating it to zero to get the value of x.
$\Rightarrow\,{f}^{'}(x)=1{e}^{x}+x{e}^{x}=0$
$0 = {e^x}(x + 1)$
We have two cases $(x + 1) = 0$ and ${e^x} = 0$
$x = - 1$ and $x = - \infty $
Now, calculating $\dfrac{{{d^2}f}}{{d{x^2}}}$ of the given function
$ \Rightarrow {f^{''}}(x) = {e^x} + {e^x} + x{e^x} = 2{e^x} + x{e^x}$
Substituting values of $x = - 1$ and $x = - \infty $ in $\dfrac{{{d^2}f}}{{d{x^2}}}$ to check point of maxima or point of minima.

First substituting $x = - 1$ we have,
\[ \Rightarrow {f^{''}}( - 1) = 2{e^{ - 1}} + ( - 1){e^{ - 1}}\]
\[ \Rightarrow {f^{''}}( - 1) = \dfrac{1}{e} > 0\]
Therefore, we can say that point $x = - 1$ is a point of minima

Hence, the correct option is D.

Note: Maxima and the minima of the function can be calculated by using a differentiation method. Care should be taken while taking the sign-in double differentiation. Also when the double derivative comes to zero it is called the inflection point (Neither minima nor maxima).