Question

In \[( - 4,4)\],the function \[f(x) = \]Integral from \[ - 10\] to \[x\] \[({t^4} - 4){e^{ - 4t}}\]dt
\[1)\] No extrema
\[2)\] One extremum
\[3)\]Two extrema
\[4)\]Four extrema

Answer 1

Hint: Before starting the question let's first know what is the extremum of a function. In calculus mathematics, any point at which the value of the given function is maximum or minimum in the given interval.
In this, the given question is exactly asking to find how many extrema on the interval \[( - 4,4)\].
First, we will differentiate the given function and then we will equate it with zero to find all the critical points of \[f(x)\]
When we equate the expression with zero then we will solve it to find the values of x.

Complete step-by-step answer:
The function is given by
\[f(x) = \int\limits_{ - 10}^x {({t^4} - 4)} {e^{ - 4t}}dt\]
Now, we need to find the derivative so we can find the critical points of \[f(x)\]. We get,\[f'(x) = f\int\limits_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} \]
As we know differentiation of integral is equal to \[1\], then we have,
\[f'(x) = 1.({x^4} - 4){e^{ - 4x}} - 0\]
$ \Rightarrow $\[f'(x) = ({x^4} - 4){e^{ - 4x}}\]
Now, put \[f'(x) = 0\], then we get,
\[({x^4} - 4){e^{ - 4x}} = 0\]
$ \Rightarrow $\[({x^4} - 4) = 0\]
$ \Rightarrow $\[{x^4} = 4\]
$ \Rightarrow $\[{x^2} = 2\]
$ \Rightarrow $\[x = \pm \sqrt 2 \]
Here we can see that there are two critical points which are found, which are \[x = + \sqrt 2 \]and \[x = - \sqrt 2 \].
So,

At \[x = + \sqrt 2 \]and \[x = - \sqrt 2 \] the given function has extreme value.
Hence option \[(3)\]is correct because our function has contained two extrema.

So, the correct answer is “Option 3”.

Note: Before finding the extremum always check that the function is continuous on the given interval or not. Because the procedure will only work if our function is continuous.
Always remember that we are only interested in what the given function is doing in the given interval, we don’t care about the critical value which falls outside the given interval.
We should always be careful about the critical points which we have to include and which we have to exclude because students always make mistakes in this situation.