The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$. We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by: $$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$ We can assert that: a. The function $f$ is convex on $\mathbb { R }$ b. The function $f$ is concave on $\mathbb { R }$ c. Point C is the unique inflection point of $\mathscr { C } _ { f }$ d. $\mathscr { C } _ { f }$ has no inflection point
The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$.\\
We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A.\\
Questions 1 to 3 relate to this same function $f$.
We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by:
$$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$
We can assert that:\\
a. The function $f$ is convex on $\mathbb { R }$\\
b. The function $f$ is concave on $\mathbb { R }$\\
c. Point C is the unique inflection point of $\mathscr { C } _ { f }$\\
d. $\mathscr { C } _ { f }$ has no inflection point