Are we expected to "think outside the book" when answering the exercises for Courant's Differential and Integral Calculus?

Question

I'm reading Courant's Differential and Integral Calculus, there is this session of exercises:

My trouble here is that previously he talks about the rational numbers, why it is needed for them to be extended, he talks a little bit about real numbers and their representation as infinite decimals, expression of numbers in scales other than $10$, inequalities and the Schwarz's Inequality. My question is: Are we supposed to prove all these using only what was given before in the book? We have two classes of exercises:

• Exercises in which it is obvious that you need what was given previously: It is quite obvious that to solve - for example - $6,8,etc$ you can use what was given in the sections about inequalities, Schwarz' inequality, etc.

• Exercises in which it is not obvious that you need what was given previously: For example, to solve $1.a$, I used the common proof with factorization of integers, $1d$ needs the test of rational roots, which was not given previously unless it was under some disguise, $2$ might need something else but I have been thinking if the Schwarz' inequality could be used to prove it, I have been doing the same (Schwarz) for the other examples without success.

This is important for me because I'm not sure what tools I am allowed to use and I know that there could be some out-of-the-box technique in which it's much easier to prove things, for example: For the examples in $5c$, I could use maxima/minima from single-variable calculus, for $5a,c$ I could use maxima/minima from multivariable calculus, but that would take all the fun from it.

Answer 1

Are we supposed to prove all these using only what was given before in the book?

Since it's a calculus book, I would assume that anything precalculus including algebra and trigonometry is fair game to use.

$1d$ needs the test of rational roots, which was not given previously

In my opinion this falls under the algebra/precalculus umbrella. Anyway, you can essentially derive the rational root theorem in much the same way you proved that $\sqrt{2}$ is irrational without explicitly using the rational root theorem for $x^2-2=0$.

$2$ might need something else but I have been thinking if the Schwarz' inequality

Remember the $\tan(\alpha-\beta)$ formula, and use that $\tan(\pi / 3)$ is irrational.

in $5c$, I could use maxima/minima from single-variable calculus

Easier: write it as $\displaystyle\,x^2\left(x^2+\frac{1}{x^2}-3\left(x+\frac{1}{x}\right)+4\right)=x^2(y^2-3y+2)\,$ where $\displaystyle\,y=x+\frac{1}{x}\,$.

for $5a,c$ I could use maxima/minima from multivariable calculus

Easier: note for example that $\displaystyle\,x^2+xy+y^2=\frac{x^3-y^3}{x-y}\,$.

Answer 2

For which problem exactly you want to see a solution?

The proof for $10$.

It's just the triangle inequality for $\Delta ABO$,

where $A(a_1,a_2,...,a_n)$, $B(b_1,b_2,...,b_n)$ and $O(0,0,...,0)$: $$AB\leq OA+OB.$$

We can prove it by C-S if you wish.

Indeed, after squaring of the both sides we need to prove that $$\sum_{i=1}^na_i^2+\sum_{i=1}^nb_i^2+2\sqrt{\sum_{i=1}^na_i^2\sum_{i=1}^nb_i^2}\geq\sum_{i=1}^n(a_i-b_i)^2$$ or $$\sqrt{\sum_{i=1}^na_i^2\sum_{i=1}^nb_i^2}\geq-\sum_{i=1}^na_ib_i,$$ which is true because $$\sqrt{\sum_{i=1}^na_i^2\sum_{i=1}^nb_i^2}\geq\sqrt{\left(\sum_{i=1}^na_ib_i\right)^2}=|\sum_{i=1}^na_ib_i|\geq-\sum_{i=1}^na_ib_i.$$ Done!