Maximum and minimum values of functions. Hello friends, today it’s about the maximum and minimum values of functions. Have a look!!

**Maximum and minimum values of functions**

Let’s say is a function. Now for any maximum or minimum value of , the value of will be zero. And if the value of is positive, then that is the minimum value of . Also, for the maximum value of , the value of will be negative.

Next, I’ll solve two examples for you.

**Examples of maximum and minimum values of functions**

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013)* “If , show that when and . Hence find the maximum and minimum values of .”

**Solution**

Now here the given function is

(1)

And I have to prove that when and . First of all, I’ll differentiate the equation (1).

**Step 1**

Since this is an implicit function, I’ll differentiate it in the same way as the * differentiation of implicit functions*.

So, I’ll differentiate equation (1) with respect to to get

Then I’ll simplify it. And that gives

(2)

(3)

Now I’ll differentiate equation (2) with respect to by using the * product rule of differentiation*. And that gives

Next, I’ll simplify it to get

(4)

Now comes my second step.

**Step 2 **

Next, I’ll prove that for and .

So, I’ll substitute in equation (3). And that gives

So that means

since can not be equal to . Thus it gives

So one of the relations is proved. Now I’ll prove the other one.

Therefore I’ll substitute and in equation (4) to get

Then I’ll simplify it. And that gives

Hence I can say that

Hence that other relation is also proved. Now I’ll find out the maximum and minimum values of .

**Step 3**

First of all, I’ll rewrite equation (1) in term of . So that gives

Next, I’ll substitute . So that gives

Thus I can say

Now for to be maximum, has to be negative and for to be minimum, has to be positive. Also, I have already proved that for and .

So I’ll substitute in . And that gives

Thus

Hence is the minimum value.

Next, I’ll substitute in . And that gives

Then I’ll simplify it to get

Thus

Hence is the maximum value.

Hence I can conclude that is the minimum and is the maximum value.

Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013)* “If , prove that and that the maximum value of occurs where and the minimum value where .”

**Solution**

Now here the given function is

(5)

And I have to prove that and that the maximum value of occurs where and the minimum value where . So I’ll start with the differentiation of equation (5).

**Step 1**

So I’ll differentiate (5) with respect to to get

Then I’ll simplify it to get the value of . Thus I get

Now I’ll cancel through out to get

(6)

So this gives

Hence I have proved the relation. Now comes my second step.

**Step 2**

Now for to be maximum or minimum, has to be . So that means

And that gives

Next, I’ll substitute in equation (5) to get

Then I’ll simplify it to get

As I can see, equation (5) is now a quadratic equation of . Now I’ll solve it for .

Since we all know the standard solution of a quadratic equation is . Now I’ll use the same formula to get the value of . Thus it will be

Then I’ll simplify it to get

So I have two values of for which the maximum and the minimum values of occur. Now I’ll check which value of is for the maximum value of and which one is for the minimum one.

**Step 3 **

Now will be minimum if is positive and the opposite if negative. So first I have to get the value of . And that I can get by differentiating the equation (6).

So I’ll differentiate the equation (6) with respect to to get

Next, I’ll simplify it to get

Now for any maximum or minimum value of . So I can rewrite this expression as

(7)

Also, from equation (6) I know that . Again from , I can say

Next, I’ll replace in equation (8) with to get

Then I’ll simplify it to get

So I can say that for , the value of is

(8)

**Step 4**

Next, I’ll get the value of for as I have got in Step 2.

So I’ll substitute in equation (8) to get

Now I’ll simplify it. And that gives

Thus I can say that the maximum value of is at .

Next, I’ll substitute in equation (8) to get

Then I’ll simplify it. And that gives

Thus I can say that the minimum value of is at .

Hence I have proved all the relations. And that is the answer to this example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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