9.3 Workshop 3 (Construction)

In this workshop, work on solving all of the following seven problems. You are encouraged to discuss the problems with those in your group but DO NOT work on writing the proofs up together.

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Your assessed problem H05 is Question 6. You must solve this problem, write up its proof and submit it on Gradescope as part of your portfolio assessment by the deadline stated below. Your write-up of the assessed proof must be your OWN work.
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More guidance on submitting and resubmitting your proofs can be found on IMU Learn $\rightarrow$ Assessment.
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You will be expected to talk and answer questions about your solutions to the remaining problems with your group in the Communication workshop next week.
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You are free to use any of the results in the summary notes in your proofs if they are relevant to your solution.

Important Dates

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DEADLINE FOR ASSESSED PROOF H05: 10:00AM Tuesday 18th November 2025
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H05 Feedback returned: 10:00AM Monday 24th November 2025
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H05 Resubmission deadline: 10:00AM Wednesday 26th November 2025 (Remember to include your reflection! And take note of the tight turnaround time to resubmit: 48 hours.)
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H05 Final deadline: 10:00AM Tuesday 2nd December 2025
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H05 RESIT DEADLINE: 10:00AM Friday 24th April 2026

9.3.1 Problems

1.

H05 Resit: Prove that

$(n+1)^{3}\leq 3n^{3}\mbox{ for all }n\geq 3.$
2.

Prove that

$(a+b)(b+c)(c+a)\geq 8abc$

for all positive $a,b,c\in\mathbb{R}$ .
3.
Prove that $(a+b)^{2}=4ab+(a-b)^{2}$ for all $a,b\in\mathbb{R}$ , and use this identity to show:
1. (a)
  
  if the sum of $a$ and $b$ is fixed then $ab$ is largest when $a=b$ .
2. (b)
  
  if the product of $a$ and $b$ is fixed and non-negative then $|a+b|$ is smallest when $a=b$ .
NOTE: These results can be proved from calculus, but for this task you must argue only using inequalities and algebra.
4.

Find all values of $x\in\mathbb{R}$ that satisfy the inequality $\displaystyle\frac{2x}{|x-1|}<1$ .
5.

Find all values of $x\in\mathbb{R}$ that satisfy $|x^{2}+3|<12$ .
6.

H05: Show that if $|x|\leq 1$ then $|x^{6}-5x-16|\geq 10$ .

Hint: Use the triangle inequalities.
7.
Let $f(n):=1/(n^{2}+1)$ .
1. (a)
  
  Assume $n\in\mathbb{R}$ , sketch the graph of $f$ .
2. (b)
  
  Assume $n\in\mathbb{N}$ , solve $|f(n)|<\frac{1}{1000}$ .
3. (c)
  
  Now let $\epsilon>0$ be any number. Solve $|f(n)|<\epsilon$ where $n\in\mathbb{N}$ .