We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$ and we call $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.
We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$. a. Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$. b. For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that: $$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
a. Determine the limit of the function $f$ at 0. b. Interpret this result graphically.
a. Determine the limit of the function $f$ at $+ \infty$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$. Draw the variation table of the function $f$ showing the limits at the boundaries of the domain of definition. c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$. a. Calculate $I$. b. Interpret the result graphically.
We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that: $$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } } .$$ a. By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.
We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$
and we call $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.
\begin{enumerate}
\item We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$.\\
a. Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$.\\
b. For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that:
$$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
\item a. Determine the limit of the function $f$ at 0.\\
b. Interpret this result graphically.
\item a. Determine the limit of the function $f$ at $+ \infty$.\\
b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$.\\
Draw the variation table of the function $f$ showing the limits at the boundaries of the domain of definition.\\
c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
\item We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$.\\
a. Calculate $I$.\\
b. Interpret the result graphically.
\item We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that:
$$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } } .$$
a. By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$.\\
b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.
\end{enumerate}