Q 3 - Calculus

Question

How do you find the critical points of a function of two variables, $f\left(x,y\right)$? What is the significance of the critical points? 

Step-by-Step Solution

Answer

The points where the first partial derivatives $f_x, f_y$ of the function $f(x,y)$ are zero are called critical points.

Step 1. Given Information

We have a function of two variables $f(x,y)$.

Then we have to explain that how we can find the critical of two variable function.

Also we have to explain that what are the significance of the critical points. 

Step 2. Explanation to find critical points

We have a two variable function $f(x,y)$.

We know that a critical point of a function exists where its first derivative is equals to zero or infinite.

In one variable function to find critical points we put the first derivative of function is equals to zero. Then we get the points that are called critical points.

But here we talk about two variable function. So we have to find partial derivatives.

That is we have to find the partial derivative of  above function with respect to $x$ and with respect to $y$.

That is we have to find $f_x$ and $f_y$.

Now we know that critical points exists where first derivative of the function is zero.

So that put these partial derivatives equals to zero.

That is :-

$f_x=0, f_y=0$.

By solving these equations we will get one or more points and that points will be critical points of the functions.

In short we can say that the points where the first partial derivatives of the function $f\left(x,y\right)$ are zero are called critical points.

Step 3. Significance of critical points

To find the critical points we take first partial derivatives of the function $f(x,y)$ equals to zero.

Let the point $\left(x,y\right)$ is a critical point of the function $f\left(x,y\right)$. That is for this point first partial derivatives are equals to zero.

That means for this point first partial derivatives does not exist.

So we can say that the points where partial derivatives of two variable function does not exists are called critical points.

We know that the derivative of a function does not exist at local extremes.

So that the local extremes of a two variable function are critical points.