31) $y= x^4 -2x^2$

$\to y' = 4x^3 - 4x$

$y' = 0 \Leftrightarrow \left[\begin{array}{l}x = -1\\x =0\\x =1\end{array}\right.$

Bảng biến thiên:

$\begin{array}{|l|cr|}
\hline
x & -\infty & & -1 & & &0 & & & 1 & & +\infty\\
\hline
y' & & - & 0& & + & 0 & - & &0& + &\\
\hline
&+\infty&&&&&0&&&&&+\infty\\
y & &\searrow& &&\nearrow & &\searrow && &\nearrow\\
&&&-1&&&&&&-1\\
\hline
\end{array}$

32) $y = \dfrac{3x+1}{x-1}$

$TXĐ: D = \Bbb R \backslash\{1\}$

$+) \quad \lim\limits_{x \to \pm}y = \lim\limits_{x \to \pm}\dfrac{3x+1}{x-1} = \lim\limits_{x \to \pm}\dfrac{3 + \dfrac{1}{x}}{1 - \dfrac{1}{x}} = 3$

$\to$ Đồ thị nhận đường thẳng $y = 3$ làm tiệm cận ngang

$+) \quad \lim\limits_{x \to 1^+}y = \lim\limits_{x \to 1^+}\dfrac{3x+1}{x-1}=+\infty$

$\qquad \lim\limits_{x \to 1^-}y = \lim\limits_{x \to 1^-}\dfrac{3x+1}{x-1}=-\infty$

$\to$ Đồ thị nhận đường thẳng $x=1$ làm tiệm cận đứng

33) Ta có:

$SA\perp (ABCD)$

$\to V_{S.ABCD} = \dfrac13S_{ABCD}.SA = \dfrac13\cdot a^2\cdot3a = a^3$

34) $+) \quad y=  4^x$

$\to y' = 4^x.\ln4$

$+) \quad y = \ln(x+1)$

$ĐKXĐ: x+1 >0 \Leftrightarrow x >-1$

$\Rightarrow TXĐ: D = (-1;+\infty)$

35) Ta có:

$l = 5a;\, h = 3a$

$\to r = \sqrt{l^2 - h^2} = \sqrt{25a^2 - 9a^2} = 4a$

$\to S_{xq} = \pi rl = \pi\cdot 4a\cdot 5a = 20a^2\pi$

36a) $2^x = 1024$

$\to 2^x = 2^{10}$

$\to x = 10$

b) $\dfrac{1}{5 - \log_2x} + \dfrac{2}{1 + \log_2x} = 1\qquad (*)$

$ĐKXĐ: \begin{cases}x >0\log_2x \ne 5\\\log_2x \ne -1\end{cases} \to \begin{cases}x >0\\x \ne 32\\x \ne \dfrac12\end{cases}$

$(*)\Leftrightarrow \dfrac{1+\log_2x + 2(5 - \log_2x)}{(5-\log_2x)(1 + \log_2x)} = 1$

$\Leftrightarrow 11 - \log_2x = (5-\log_2x)(1 + \log_2x)$

$\Leftrightarrow 11 - \log_2x = 5 +4\log_2x - \log_2^2x$

$\Leftrightarrow \log_2^2x - 5\log_2x + 6 = 0$

$\Leftrightarrow \left[\begin{array}{l}\log_2x = 2\\\log_2x = 3\end{array}\right.$

$\Leftrightarrow \left[\begin{array}{l}x = 4\\x = 8\end{array}\right.\quad (nhận)$

Vậy $S=\{2;8\}$