Proving inequalities
Prove algebraic inequalities using the identity for all real ; apply the AM-GM inequality in two and three variables; prove and apply inequalities involving two or more variables.
Worked examples
Basic inequality from a perfect square
Straightforward
Problem
Prove that for all real , .
1
Rearrange the inequality to show LHS RHS .
It suffices to show .
2
Express the left side as a perfect square.
.
3
Conclude using the non-negativity of squares.
Since for all real , , the inequality holds, with equality iff .
Answer
for all real , with equality iff .
Proving AM–GM for two variables
Moderate
Problem
Prove that for all non-negative , , .
1
Substitute , and use the non-negativity of a square.
Let , (both ). Then .
2
Expand and rewrite in terms of and .
.
3
Rearrange to obtain the AM-GM inequality.
. Equality holds iff , i.e. .
Answer
for non-negative , with equality iff .
Applying AM–GM to find an extremum
Challenging
Problem
Find the minimum value of for .
1
Apply AM-GM to the two positive terms and .
.
2
Multiply through by 2 to obtain the inequality.
.
3
Determine when equality holds and verify.
Equality holds when , i.e. , i.e. (taking ). Verification: .
Answer
The minimum value is , achieved at .
Practise
Q1·Straightforward
Prove that for all real numbers and .
State the condition for equality.
State the condition for equality.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
For all real numbers and :
(A square of a real number is always non-negative.)
Expand the left side:
Add to both sides:
**Equality condition:** if and only if . So equality holds if and only if .
**Remark:** This is equivalent to the AM-GM inequality for two variables: , or equivalently when (after substituting , ).
For all real numbers and :
(A square of a real number is always non-negative.)
Expand the left side:
Add to both sides:
**Equality condition:** if and only if . So equality holds if and only if .
**Remark:** This is equivalent to the AM-GM inequality for two variables: , or equivalently when (after substituting , ).
Q2·Straightforward
Prove that for all real numbers , and hence prove that .
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
Observe that:
For all real , .
Therefore:
Add to both sides:
**Equality condition:** iff . So equality holds iff .
**Check:** When : LHS , RHS . ✓
Observe that:
For all real , .
Therefore:
Add to both sides:
**Equality condition:** iff . So equality holds iff .
**Check:** When : LHS , RHS . ✓
Q3·Straightforward
The AM-GM inequality for two non-negative numbers and states:
Explanation
**Option (b)** is correct: for all non-negative reals and ,
with equality if and only if .
The left side is the **arithmetic mean** (AM) and the right side is the **geometric mean** (GM).
**Quick verification:** Let , .
- AM
- GM
- ✓
The equality case: .
- AM , GM . ✓
with equality if and only if .
The left side is the **arithmetic mean** (AM) and the right side is the **geometric mean** (GM).
**Quick verification:** Let , .
- AM
- GM
- ✓
The equality case: .
- AM , GM . ✓
Q4·Moderate
Prove the AM-GM inequality: for all non-negative real numbers and ,
and state when equality holds.
and state when equality holds.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
Since , the square roots and are real. Let and .
For all real and :
Substitute back , , :
Divide by 2:
**Equality condition:** iff iff iff .
So equality holds if and only if .
Since , the square roots and are real. Let and .
For all real and :
Substitute back , , :
Divide by 2:
**Equality condition:** iff iff iff .
So equality holds if and only if .
Q5·Moderate
Prove that for all real numbers , , :
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
For all real numbers, any square is non-negative:
Add all three inequalities:
Divide by 2:
Therefore:
**Equality condition:** All three squares , , must equal zero simultaneously, which requires .
For all real numbers, any square is non-negative:
Add all three inequalities:
Divide by 2:
Therefore:
**Equality condition:** All three squares , , must equal zero simultaneously, which requires .
Q6·Moderate
Prove that for all positive real numbers and :
and state when equality holds.
and state when equality holds.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof (Method 1 — clear denominators):**
Since , we have . Multiply both sides of by :
This is equivalent to , which is always true.
Since all steps are reversible (we multiplied by a positive quantity), the original inequality holds.
**Proof (Method 2 — AM-GM directly):**
Let and . Note and .
By AM-GM: .
Therefore .
**Equality condition:** .
Since , we have . Multiply both sides of by :
This is equivalent to , which is always true.
Since all steps are reversible (we multiplied by a positive quantity), the original inequality holds.
**Proof (Method 2 — AM-GM directly):**
Let and . Note and .
By AM-GM: .
Therefore .
**Equality condition:** .
Q7·Moderate
Given that and are positive real numbers with , use the AM-GM inequality to find the maximum possible value of .
Explanation
Apply the AM-GM inequality to and :
Substitute :
Square both sides (both sides are non-negative):
The maximum value of is .
**When is equality achieved?** AM-GM gives equality iff . With and , we get . Check: . ✓
Substitute :
Square both sides (both sides are non-negative):
The maximum value of is .
**When is equality achieved?** AM-GM gives equality iff . With and , we get . Check: . ✓
Q8·Moderate
Given that and are positive real numbers with , find the maximum possible value of using the AM-GM inequality.
Explanation
By AM-GM:
With :
The maximum value of is , achieved when .
**Verification:** ✓, and ✓.
With :
The maximum value of is , achieved when .
**Verification:** ✓, and ✓.
Q9·Challenging
Prove that for all real numbers and :
Also state the condition for equality.
Also state the condition for equality.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
We need to show , i.e., .
Expand:
Since for all real :
**Equality condition:** iff .
**Application:** This inequality arises in statistics — the variance of two numbers satisfies , and this inequality reflects the same idea.
We need to show , i.e., .
Expand:
Since for all real :
**Equality condition:** iff .
**Application:** This inequality arises in statistics — the variance of two numbers satisfies , and this inequality reflects the same idea.
Q10·Challenging
Prove the Cauchy-Schwarz inequality for two pairs of real numbers:
for all real .
for all real .
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Proof:**
Consider the expression .
Expand:
Subtract:
Since :
**Equality condition:** , i.e., (when ) — the pairs are proportional.
Consider the expression .
Expand:
Subtract:
Since :
**Equality condition:** , i.e., (when ) — the pairs are proportional.
Q11·Challenging
For , find the minimum value of using the AM-GM inequality.
Explanation
Apply AM-GM to and (both positive since ):
Therefore:
The minimum value is .
**When is equality achieved?** AM-GM gives equality when , i.e., , so (since ).
**Verification:** ✓.
Therefore:
The minimum value is .
**When is equality achieved?** AM-GM gives equality when , i.e., , so (since ).
**Verification:** ✓.
Q12·Challenging
Given with , find the maximum value of .
Explanation
By AM-GM:
With :
The maximum value of is , achieved when .
**Verification:** ✓ and ✓.
**Geometric interpretation:** Among all rectangles with a fixed perimeter, the square has the largest area. Here the 'perimeter' condition is , and the 'area' is .
With :
The maximum value of is , achieved when .
**Verification:** ✓ and ✓.
**Geometric interpretation:** Among all rectangles with a fixed perimeter, the square has the largest area. Here the 'perimeter' condition is , and the 'area' is .
Open Math
Proving inequalities
Proof · MEX-12-01
Name:
Date:
Q1Straightforward
Prove that for all real numbers and .
State the condition for equality.
State the condition for equality.
Q2Straightforward
Prove that for all real numbers , and hence prove that .
Q3Straightforward
The AM-GM inequality for two non-negative numbers and states:
- A.
- B., with equality iff
- C., with equality iff
- D., with equality iff
Q4Moderate
Prove the AM-GM inequality: for all non-negative real numbers and ,
and state when equality holds.
and state when equality holds.
Q5Moderate
Prove that for all real numbers , , :
Q6Moderate
Prove that for all positive real numbers and :
and state when equality holds.
and state when equality holds.
Q7Moderate
Given that and are positive real numbers with , use the AM-GM inequality to find the maximum possible value of .
Q8Moderate
Given that and are positive real numbers with , find the maximum possible value of using the AM-GM inequality.
Q9Challenging
Prove that for all real numbers and :
Also state the condition for equality.
Also state the condition for equality.
Q10Challenging
Prove the Cauchy-Schwarz inequality for two pairs of real numbers:
for all real .
for all real .
Q11Challenging
For , find the minimum value of using the AM-GM inequality.
Q12Challenging
Given with , find the maximum value of .
Worked solutions and answers at openmath.au/year-12/extension-2/the-nature-of-proof/proving-inequalities