The image contains a math exercise in French that involves several tasks based on a given graph and function.

Let's break down the questions in the image:

1) The graph in question represents an odd function $h$ defined over the interval $[0,4]$.

a) Determine $D_h$, the domain of definition of $h$:

• The graph is drawn for the interval $[0,4]$. The domain of the function is $D_h = [0,4]$, which means the function is defined for all values between 0 and 4, inclusive.

b) Complete the curve representing the function $h$:

• To complete the curve, we would need more information about the function's behavior on the graph. From the graph, you can identify key points and smooth transitions based on the function's pattern. Typically, odd functions are symmetric about the origin, so if it's known that the function is odd, you can mirror the part shown on the positive side onto the negative side.

c) Determine the variations of the function $h$:

• The variations of the function involve analyzing how the function increases or decreases within the interval. You would need to check the critical points (where the derivative equals zero) and analyze the intervals where the function is increasing or decreasing.

d) Show that $h$ is bounded:

• A function is bounded if it doesn't go to infinity in either direction. By examining the graph, you can determine whether the function's values are contained within a certain range. If the graph shows that $h(x)$ remains within a specific range for all $x \in [0,4]$, then the function is bounded.

e) Determine the sign of $h$:

• The sign of $h$ tells us whether the function is positive or negative over its domain. From the graph, identify whether the function is above or below the x-axis. If the curve lies entirely above the x-axis, then $h(x)$ is positive; if it lies below, then $h(x)$ is negative.

f) Determine the expression of the function $h$:

• To determine the expression of the function, you would need to derive it from the graph. For an odd function, the general form could be something like $h(x) = ax^3 + bx^2 + cx + d$, and you would use the known points to find the coefficients.

2) Given two real numbers $P(x, y) = x^2 + y^2$, you need to show whether $P(x, y)$ has a maximum value at the point $(x, y) = (0, 0)$:

• To show this, you would evaluate $P(x, y)$ at the given point and check if there are any other points within the domain that give a larger value. The function $P(x, y) = x^2 + y^2$ represents a circular paraboloid, and it achieves its minimum value (not maximum) at $(0, 0)$, where $P(0, 0) = 0$. Since the function increases as $x$ and $y$ move away from the origin, this is indeed a minimum, not a maximum.

Summary of steps for solution:

• Determine the domain of the function.
• Complete the curve based on symmetry and given points.
• Analyze the behavior of the function (increasing or decreasing).
• Verify boundedness by observing the range on the graph.
• Determine the sign of the function based on whether it's above or below the x-axis.
• Derive the function's formula from the graph, potentially using odd function characteristics.
• For $P(x, y)$, check if the point $(0, 0)$ corresponds to a minimum or maximum, and analyze the function's behavior at other points.

Would you like further explanations on any of these steps? Or is there a specific part you'd like me to focus on?