Question

What is the nature of the graph:
$y=6{{e}^{-4x}}$
A. Monotonically Increasing
B. Monotonically Decreasing
C. Increasing then Decreasing
D. Decreasing then Increasing

Answer 1

Hint : Here we have to find the nature of the graph whether it is monotonically decreasing or increasing. We will use derivatives to find the nature of the graph. Firstly we will find the derivative of the function given then we will check for all values of $x$ whether the function is increasing or decreasing and get our desired answer.

Complete step-by-step answer :
The exponential function is given as below:
$y=6{{e}^{-4x}}$
The graph of the function can be drawn on a $x-y$ plane as follows:

For finding whether the graph is increasing or decreasing we will firstly find its derivative as follows:
$\dfrac{dy}{dx}=6\left( \dfrac{d\left( {{e}^{-4x}} \right)}{dx} \right)$ ……$\left( 1 \right)$
The differentiation of the exponential term is calculated by the below formula:
$\dfrac{d\left( {{e}^{ax}} \right)}{dx}=a{{e}^{ax}}$
Where $a\in R$
So using above formula in equation (1) we get,
$\dfrac{dy}{dx}=6\left( \left( -4 \right){{e}^{-4x}} \right)$
$\Rightarrow \dfrac{dy}{dx}=-24{{e}^{-4x}}$
So as we can see that for any $x\in R$
$\dfrac{dy}{dx}=-24{{e}^{-4x}}<0$
So the function is exponentially decreasing.
Hence it is a monotonically decreasing exponential function.
Hence option (B) is correct.
So, the correct answer is “Option B”.

Note : Exponential functions are used to denote whether there is a growth in a function or decay depending upon the graph of the function. In exponential growth the graph increases very slowly at first and then rapidly similarly in exponential decay the graph decreases rapidly at first and then slowly. The Exponential function general form is given as $f\left( x \right)={{b}^{x}}$ where $b$ is any positive number which is always greater than $1$ and $x$ is the exponent of the function. The most commonly used exponential function is $f\left( x \right)={{e}^{x}}$ which is also known as a natural exponential function. In natural exponential function $e$ is a Euler’s number and its value lies between $2\And 3$ .