# Mathematics Extension 2 – Harder Inequalities

Harder Inequalities

## Algebraic Inequalities

In the extension 1 course, we have looked at solving simple inequalities with the unknown on the denominator and proving basic inequalities using Induction. This topic is an extension on the extension 1 course where we prove inequalities using algebra, calculus and Mathematical Induction. However, we will cover inequalities involving Mathematical Induction later on in the Induction topic. Finally, we will look at solving inequalities using algebraic and graphical methods.

### Inequalities Involving $Latex formula$

Using the fact that the square of a real number is always greater than or equal to zero allows us to solve more complicated inequalities. As a simple exercise using $Latex formula$, let’s prove the following relationships where $Latex formula$:

(i) $Latex formula$

(ii) $Latex formula$

#### Proof

i)

$Latex formula$

$Latex formula$

$Latex formula$

ii)

From i) $Latex formula$

$Latex formula$

$Latex formula$

The methods of obtaining many inequality relationships involve:

• Expansion and simplification
• Substitution of an expression into an existing relationship
• Rearrangement
• Work forward
• Work backward

The following examples demonstrate how these concepts can be used in establishing inequality relationships.

### Example 1

Prove that $Latex formula$, where $Latex formula$ and $Latex formula$.

### Solution 1

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

### Example 2

Use the expression $Latex formula$, show that

(i) $Latex formula$

(ii) $Latex formula$

### Solution 2

i)

$Latex formula$

Similarly, $Latex formula$
and $Latex formula$
Summing up the LHS and RHS, we have
$Latex formula$

$Latex formula$

ii)

Given $Latex formula$
Let $Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

$Latex formula$

$Latex formula$

Now, we can substitute $Latex formula$ for $Latex formula$, and similarly for $Latex formula$ without loss of generality.

$Latex formula$

### Example 3

Prove that $Latex formula$, where $Latex formula$.

### Solution 3

Working backward,

$Latex formula$
<–>$Latex formula$
<–>$Latex formula$
<–>$Latex formula$

<–>$Latex formula$

<–>$Latex formula$

Which is clearly true, $Latex formula$

### Example 4

Prove that $Latex formula$, where$Latex formula$, where $Latex formula$

### Solution 4

$Latex formula$
Recall that $Latex formula$
Therefore, substituting $Latex formula$ for $Latex formula$
We have $Latex formula$

$Latex formula$

### Example 5

[HSC 2011, Question 5]

If $Latex formula$ and $Latex formula$ are positive real numbers and $Latex formula$, prove that $Latex formula$

### Solution 5

$Latex formula$

$Latex formula$

Consider the numerator:

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Numerator is greater than 0.

Since the denominator is also greater than 0, therefore $Latex formula$.

### Example 6

[HSC 2012, Question 15]

(i) Prove that $Latex formula$, where $Latex formula$ and $Latex formula$.

(ii) If $Latex formula$, show that $Latex formula$.

(iii) Let $Latex formula$ and $Latex formula$ be positive integers with $Latex formula$. Prove that $Latex formula$

(iv) For integers, prove that $Latex formula$

### Solution 6

i)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

ii)

$Latex formula$

Noting that $Latex formula$ $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

iii)

Consider $Latex formula$
Let $Latex formula$

$Latex formula$

$Latex formula$
Finally, consider $Latex formula$
To maximise this, we solve for $Latex formula$

$Latex formula$

$Latex formula$
Test: $Latex formula$ Therefore, it is a maximum.

Thus, maximum value for $Latex formula$ $Latex formula$But $Latex formula$
Thus, $Latex formula$

$Latex formula$

iv)

From iii), we have $Latex formula$ for $Latex formula$
Substituting $Latex formula$ we have

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Multiplying both sides, we get

$Latex formula$

$Latex formula$

## Further Inequalities

One of the inequalities that we have looked at earlier $Latex formula$ is a famous inequality which states the geometric mean of two positive numbers is always less than or equal to the arithmetic mean. The inequality can also be generalised for terms and gives:

$Latex formula$

Note: Students are not allowed to assume the above inequality in examinations.

In fact, we can extend the above inequality further into the following inequality.

The Harmonic Mean, Geometric Mean, Arithmetic Mean and Quadratic Mean Inequalities:

$Latex formula$

Another famous inequality is the Cauchy-Schwartz Inequality which states:

$Latex formula$

where, $Latex formula$

Note: these two above inequalities are important to know and understand but they are not mentioned in the syllabus thus students are not required to learn these.

## Application of Calculus in Inequalities

### Differential Calculus

Differential calculus can be applied to prove inequalities that require us to show whether a function is increasing or decreasing on a given interval.

If the question requires us to show $Latex formula$ for $Latex formula$, then we can follow the steps below:

1. prove that $Latex formula$, which means $Latex formula$ is increasing for $Latex formula$.
2. hence, $Latex formula$.
3. show that, if $Latex formula$, then $Latex formula$ for $Latex formula$.

and vice versa for decreasing functions.

Note: The above method can also be applied for second and higher derivatives if the first derivative is insufficient to prove $Latex formula$ for $Latex formula$.

### Integral Calculus

Integral calculus is the other branch of calculus that can be applied to prove inequalities and of course, on the bases that we are able to integrate the inequalities. Now consider the following diagram,

We can see that, $Latex formula$ on the interval $Latex formula$ and therefore,

$Latex formula$

Note: In many inequality questions, integral calculus is often used in conjunction with differential calculus.

Let’s now look at some examples!

### Example 3

[HSC 2007, Question 7]

(i) Show that $Latex formula$ for $Latex formula$.

(ii) Let $Latex formula$. Show that the graph of $Latex formula$ is concave up for $Latex formula$.

(iii) By considering the first two derivatives of $Latex formula$, show that $Latex formula$ for $Latex formula$.

### Solution 3

i)

Consider $Latex formula$

$Latex formula$

Also, $Latex formula$

Thus, $Latex formula$ is always decreasing and is equal to 0 for $Latex formula$.

Therefore, for $Latex formula$, $Latex formula$.

$Latex formula$

$Latex formula$

ii)

$Latex formula$

$Latex formula$

From i), we know that $Latex formula$ for $Latex formula$.

$Latex formula$, and is concave up.

iii)

$Latex formula$

Consider $Latex formula$

Since $Latex formula$, that means $Latex formula$ is always increasing.

Furthermore, $Latex formula$

Thus, $Latex formula$ for $Latex formula$ since it starts at 0 and is increasing.

Therefore, since $Latex formula$ and $Latex formula$ for $Latex formula$, that means $Latex formula$.

$Latex formula$ for$Latex formula$.

### Example 4

[HSC 2009, Question 8]

Let $Latex formula$ be a positive integer greater than 1.

The area of the region under the curve $Latex formula$ from $Latex formula$ to $Latex formula$ is between the areas of two rectangles, as shown in the diagram.

Show that

$Latex formula$

### Solution 4

Note that the exact area under the graph from n-1 to n is greater than the area of the smaller rectangle, and smaller than the area of the large rectangle.

Thus,

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

### Example 5

[HSC 2006, Question 8]

Suppose $Latex formula$.

(i) Show that $Latex formula$.

(ii) Hence show that $Latex formula$.

(iii) By integrating the expressions in the inequality in part (ii) with respect to $Latex formula$ from $Latex formula$ and $Latex formula$(where $Latex formula$), show that

$Latex formula$

(iv) Hence show that for $Latex formula$

$Latex formula$

### Solution 5

i)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Now,$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$
(noting that $Latex formula$)
Also, $Latex formula$
$Latex formula$.

ii)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

iii)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

iv)

Raise iii) to the power of $Latex formula$
$Latex formula$
$Latex formula$

## Solving Inequalities

We have studied solving inequations in the Maths Extension 1 course such as $Latex formula$. In the Extension 2 course, the problems will be a lot more difficult and so to solve them more efficiently, we will be using both the algebraic and graphical methods.

The following examples demonstrate how we can use both the algebraic and graphical methods to solve harder inequations.

### Example 6

Solve the following inequality for $Latex formula$,

$Latex formula$

### Solution 6

Since $Latex formula$, we can multiply both sides without changing sign (noting also that our final answer is now restricted to $Latex formula$).

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Below is the plot the cubic on the LHS:

Thus, the inequality holds when $Latex formula$ or $Latex formula$.

However, recall that $Latex formula$

Thus, solutions is $Latex formula$.

### Example 7

Solve the following inequality for $Latex formula$,

$Latex formula$

### Solution 7

$Latex formula$

Consider two cases:

Case 1: $Latex formula$

When this is true, the numerator will be positive. Thus, for the LHS to be positive, we only require that the denominator also be positive.

Solving $Latex formula$, we have:

$Latex formula$

But, $Latex formula$ is only true when $Latex formula$ or $Latex formula$

Thus, combining these, we have $Latex formula$ or $Latex formula$

Case 2: $Latex formula$

When this is true, the numerator will be negative. Thus for the LHS to be positive, we only require the denominator to be negative.

Solving $Latex formula$, we have:
$Latex formula$ or $Latex formula$

But, $Latex formula$ is only true when $Latex formula$

Thus, combining these, we have no solution for this case.

Therefore, the overall solutions is $Latex formula$ or $Latex formula$.