Mathematics Extension 2 – Harder Inequalities

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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
  • Addition and multiplication
  • 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, whereLatex 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 formulaBut 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,

integral caculus of calculus in inequalities

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 forLatex formula.

Example 4

[HSC 2009, Question 8]

hsc two thousand nine question eight

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:

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.

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