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The following four problems appeared in (UCLA) (University of California, Los Angeles) - GRE Preparation.

$\mathbf{Problem} \space \mathbf{11}, \mathbf{Problem} \space \mathbf{12}, \mathbf{Problem} \space \mathbf{13},\mathbf{and \space} \mathbf{Problem} \space \mathbf{14}$

I am not asking how to solve them individually. However, they all appeared in the first section (CALCULUS I), and I am thinking what do their solutions have in common. In other words, I am asking what is the topic in CALCULUS I have to refer.

Which of the following shows the numbers $27^{30}, 5^{40}, 4^{60}$ in increasing order?

$\text{(A)} \space 27^{30} < 5^{40} < 4^{60} \space \space \space \space \text{(B)} \space 27^{30} < 4^{60} < 5^{40} \space \space \space \space \text{(C)} \space 4^{60} < 27^{30} < 5^{40}$

$\text{(D)} \space 4^{60} < 5^{40} < 27^{30} \space \space \space \space \color{blue}{\text{(E)} \space 5^{40} < 4^{60} < 27^{30}}$

Which of the following shows the numbers $10^{1/9}, 9^{1/10}, 8^{1/11}$ in increasing order?

$\text{(A)} \space 10^{1/9} < 9^{1/10} < 8^{1/11} \space \space \space \space \text{(B)} \space 10^{1/9} < 8^{1/11} < 9^{1/10} \space \space \space \space \text{(C)} \space 8^{1/11} < 10^{1/9} < 9^{1/10}$

$\color{blue}{\text{(D)} \space 8^{1/11} < 9^{1/10} < 10^{1/9}} \space \space \space \space \text{(E)} \space 9^{1/10} < 8^{1/11} < 10^{1/9}$

Which of the following shows the numbers $e^{\pi}, e^{e}, \pi^{e}$ in increasing order?

$\text{(A)} \space e^{\pi} < e^{e} < \pi^{e} \space \space \space \space \text{(B)} \space e^{e} < e^{\pi} < \pi^{e} \space \space \space \space \color{blue}{\text{(C)} \space e^{e} < \pi^{e} < e^{\pi}}$

$\text{(D)} \space \pi^{e} < e^{e} < e^{\pi} \space \space \space \space \text{(E)} \space \pi^{e} < e^{\pi} < e^{e}$

Which of the following shows the numbers $200!, 100^{200}, 200^{100}$ in increasing order?

$\text{(A)} \space 200! < 100^{200} < 200^{100} \space \space \space \space \text{(B)} \space 200! < 200^{100} < 100^{200} \space \space \space \space \color{blue}{\text{(C)} \space 200^{100} < 200! < 100^{200}}$

$\text{(D)} \space 200^{100} < 100^{200} < 200! \space \space \space \space \text{(E)} \space 100^{200} < 200^{100} < 200!$

For the third problem, we know that $e^{\pi} > e^{e}$, so we exclude options $\text{(A)}$ and $\text{(E)}$. Also we know that $e^{\pi} > \pi^{e}$, so we exclude options $\text{(B)}$ and $\text{(D)}$. Hence $\text{(C)}$ is correct.

That is by managing the expressions to have the form $y = x^{1/x}$ then differentiating and finding the critical number(s).

Still this problem matches the others, and they might come under one topic of calculus.

Your help would be appreciated.

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  • $\begingroup$ In some way or other, all these exercises have been solved here. See for example this post. They have in common, that we should find bounds, and do not use computers. $\endgroup$
    – Dietrich Burde
    Jul 1, 2022 at 13:06
  • $\begingroup$ @DietrichBurde The question is; which topic in calculus have I to refer? $\endgroup$
    – Hussain-Alqatari
    Jul 1, 2022 at 13:10
  • $\begingroup$ I think calculus in general, i.e., estimates, differentiation, binomial expansion, Stirling formula etc. See for example the answers here, what methods are used. $\endgroup$
    – Dietrich Burde
    Jul 1, 2022 at 13:12
  • $\begingroup$ You can check this link: arxiv.org/pdf/1806.03163.pdf $\endgroup$
    – insipidintegrator
    Jul 1, 2022 at 14:52
  • $\begingroup$ They don't seem to have much in common. Obviously $0<e<\pi$ implies $e^e<\pi^e.$ And $\pi^e<e^{\pi}\iff \frac {\ln\pi}{\pi}<\frac {\ln e}{e}$ which means we should examine $f'(x)$ for $f(x)=\frac {\ln x}{x}.$ On the other hand $5^{40}<40^{60}\iff (5^{40})^{1/40}<(40^{60})^{1/40} \iff 5<4^{3/2}.$ $\endgroup$
    – DanielWainfleet
    Jul 1, 2022 at 15:50

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