Problem Bank

Determine the curve represented by the equation

and sketch it, justifying its shape.

If the function $f(x)=|x+1|+2|x-a|$ has minimum value $5$, find $a$.

Let $x_1$ and $x_2$ be the solutions of

respectively. Find $x_1x_2$.

In a regular hexagonal prism $ABCDEF\text{-}A_1B_1C_1D_1E_1F_1$ (Fig. 1) the base edge length is $1$ and the la…

Let $f(x) = \ln(1+x) - \dfrac{x(1 + \lambda x)}{1+x}$.

1. If $f(x) \le 0$ for all $x \ge 0$, find the minimum value of $\lambda$. 2. Let $a_n = 1 + \dfrac12 + \dfrac13 + \cdots + …

Let $f(x) = x\ln x$.

1. Find the extrema of $g(x) = \dfrac{f(x)}{x^2} - \dfrac1x$. 2. Prove that

Let $P$ be the solution set of the inequality

If $(0, 1] \subseteq P$, find the range of possible values of the real number $a$.

The triangle $ABC$ has angles $A, B, C$ with $C \geqslant \dfrac{\pi}{3}$, and sides $a, b, c$. Prove that

A sequence $\{a_n\}$ of positive terms has partial sums $S_n$. If both $\{a_n\}$ and $\left\{\sqrt{S_n}\right\}$ are arithmetic sequences with the same common difference $d$, find …

Let $f(x) = \left(1 + ax^2\right)\mathrm{e}^x - 1$.

1. For $a \geqslant 0$, discuss the monotonicity of $f(x)$. 2. Find the number of zeros of $f(x)$ on $[0, 1]$.

For every positive integer $n$ and real $\theta$, prove that

A sequence $\{a_n\}$ satisfies

Find