Question

How do you find extreme values of a function and where they occur?

Answer 1

Hint: In the given question, we have been asked how we can calculate the extreme values of a function, or it can be said that how we can calculate the absolute maximum and the absolute minimum of a function. To solve this question, we take an arbitrary function (as we have to tell the procedure in general), then we apply the formula by writing the procedural steps and the general solutions too.

Complete step by step answer:
The derivative of a function is calculated using-
\[f'\left( {{x^n}} \right) = n{x^{n - 1}}\]
In the given question, we are asked how we can calculate the absolute maximum and the absolute minimum of a function.
Consider we have a function, \[f\left( x \right)\].
First, we find the derivative of \[f\left( x \right)\].
The derivative is, \[f'\left( x \right)\].
Now, we find the points where the derivative equals zero,
\[f'\left( x \right) = 0\]
Let the points be \[{x_1},{x_2},...,{x_n}\].
Finally, we put in the points \[{x_1},{x_2},...,{x_n}\] in the original function, \[f\left( x \right)\] and find the value at each point, i.e., we find
\[f\left( {{x_1}} \right),f\left( {{x_2}} \right),...,f\left( {{x_n}} \right)\]
And we then pick up the value where \[f\left( {{x_m}} \right)\] gives the maximum and minimum value for the function and the corresponding argument gives us the answer.

Note: In the given question, we were asked how we can find extreme values of a function. We solve these equations by finding the zeros of the derivative and then putting the value of the zeroes in the original function. Some students make mistakes by reversing the order of finding the zeros and putting the value – find the zeros of the original function and put the value in the equation of the derivative. So, care must be taken at that point.