Given the function $f ( x ) = x ^ { 3 } - k x + k ^ { 2 }$ .
(1) Discuss the monotonicity of $f ( x )$;
(2) If $f ( x )$ has three zeros, find the range of values for $k$ .

Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point.
(1) If $x _ { 1 } = - 1$ , find $a$ ;
(2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .

Given the function $f ( x ) = a x - \frac { 1 } { x } - ( a + 1 ) \ln x$ .
(1) When $a = 0$ , find the maximum value of $f ( x )$ ;
(2) If $f ( x )$ has exactly one zero point, find the range of values for $a$ .

Given the function $f ( x ) = \frac { e ^ { x } } { x } - \ln x + x - a$.
(1) If $f ( x ) \geq 0$, find the range of values for $a$;
(2) Prove that if $f ( x )$ has two zeros $x _ { 1 }$ and $x _ { 2 }$, then $x _ { 1 } x _ { 2 } < 1$.

Given that the function $f(x)=a\mathrm{e}^x-\ln x$ is monotonically increasing on the interval $(1,2)$, the minimum value of $a$ is
A. $\mathrm{e}^2$
B. $\mathrm{e}$
C. $\mathrm{e}^{-1}$
D. $\mathrm{e}^{-2}$

Let $f ( x ) = ( x + a ) \ln ( x + b )$. If $f ( x ) \geq 0$, then the minimum value of $a ^ { 2 } + b ^ { 2 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 2 }$
D. 1

Given function $f ( x ) = \mathrm { e } ^ { x } - a x - a ^ { 3 }$.
(1) When $a = 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$;
(2) If $f ( x )$ has a local minimum value that is negative, find the range of values for $a$.

(17 points) Given function $f ( x ) = \ln \frac { x } { 2 - x } + a x + b ( x - 1 ) ^ { 3 }$ .
(1) If $b = 0$ and $f ^ { \prime } ( x ) \geqslant 0$ , find the minimum value of $a$ ;
(2) Prove that the curve $y = f ( x )$ is centrally symmetric;
(3) If $f ( x ) > - 2$ if and only if $1 < x < 2$ , find the range of $b$ .

Given $f ( x ) = x + k \ln ( 1 + x )$, the tangent line to the curve at point $( t , f ( t ) ) ( t > 0 )$ is $l$.
(1) If the slope of tangent line $l$ is $k = - 1$, find the monotonic intervals of $f ( x )$;
(2) Prove that tangent line $l$ does not pass through $( 0,0 )$;
(3) Given $A ( t , f ( t ) ) , C ( 0 , f ( t ) ) , O ( 0,0 )$, where $t > 0$, and tangent line $l$ intersects the $y$-axis at point $B$. When $2 S _ { \triangle ACO } = 15 S _ { \triangle ABC }$, how many points $A$ satisfy the condition? (Reference data: $1.09 < \ln 3 < 1.10, 1.60 < \ln 5 < 1.61, 1.94 < \ln 7 < 1.95$.)

Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$.
(1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$;
(2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively.
(i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$;
(ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.

Let $U$ be the open set of $\mathbb{R}^{2}$ defined by $$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$ and let $f$ be the function defined on $U$ by $$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$
(a) Show that for $(t, x) \in U$, we have $$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$
(b) For $x > 0$, show that we have $$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$ For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f: U \to \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Let $x_0 \in \bar{U}$ be a point where $f$ attains its maximum. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

Study the convexity of the function $t \mapsto \ln(1 + \mathrm{e}^t)$.

Study the convexity of the function $t \mapsto \ln \left( 1 + \mathrm { e } ^ { t } \right)$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$, and let $x_0 \in B(a,r)$ be a point where the application $x \mapsto \left\|y_0 - f(x)\right\|^2$ attains its minimum on $\overline{B(a,r)}$.
Show that $f(x_0) = y_0$.