Problem Bank

Let $f(x) = 2ax + \mathrm{e}^x$ and $g(x) = ax^2 - 2ax - x\mathrm{e}^x$, with $a \in \mathbb{R}$.

1. Discuss the monotonic intervals of $f$. 2. If $f(x) + g(x) \le 1$ for all real…

As shown in the figure, from a point $P$ outside an ellipse, the two tangent lines $PA$ and $PB$ are drawn, touching …

Let $V_1$ be the volume of the solid obtained by rotating the region bounded by

about the $y$-axis, and $V_2$ the volume …

The parabola $C:y^2=2px$ ($p>0$) has focus $F$ and directrix $l:x=-1$. Points $A(x_1,y_1)$ and $B(x_2,y_2)$ (with $y_2>y_1\geqslant 0$) lie on $C$, the line $AB$ meets $l$ at a poi…

Suppose that for every $x \in [1, +\infty)$,

Determine, with proof, which of the following are true:

1. …

The function

has a minimum value. Find the range of $a$.

Real numbers $a, b$ satisfy $a^2 \geqslant 4b$. Find the minimum value of

A function $f$ defined on $\mathbb{R}$ satisfies

and $f(x_1) \leqslant f(x_2)$ whenever $0 \leqslant x_1 \leq…

Let $M$ be the largest of the four numbers

Given that $c + d \geqslant 2$, find the minimum possible value of $M$.

Determine, with proof, which of the following are true:

1. $1.9^2 < 2^{1.9}$; 2. $2^{2.9} < 2.9^2$; 3. $\log_7 4 < \log_{12} 7$; 4. $\log_7 4 + \log_{12} 7 < \sqrt{2}$.

A television series has $n+1$ episodes, where $n\geqslant 10$. After watching the first episode, a viewer plans the rest as follows:

1. counting days from the day after finishing …

A triangular lawn $\triangle ABC$ has side lengths $30$, $40$, and $50$ meters. Points $D, E$ lie on the sides of the triangle, and the segment $DE$ divides the lawn into two parts…