This problem involves analyzing the function $s(x) = x^3 - \frac{33}{4} x^2 - \frac{63}{2} x + \frac{1}{3}$ defined on the interval $[0; 10]$.

Here is the breakdown of the tasks:

a. Derivative of the function $s(x)$

To find the expression of the derivative $s'(x)$, we apply standard differentiation rules (power rule for each term):

$s'(x) = \frac{d}{dx} \left( x^3 - \frac{33}{4} x^2 - \frac{63}{2} x + \frac{1}{3} \right)$

Let's calculate this derivative term by term:

• Derivative of $x^3$ is $3x^2$
• Derivative of $-\frac{33}{4}x^2$ is $-\frac{33}{2}x$
• Derivative of $-\frac{63}{2}x$ is $-\frac{63}{2}$
• Derivative of the constant $\frac{1}{3}$ is $0$

So, the derivative $s'(x)$ is:

$s'(x) = 3x^2 - \frac{33}{2}x - \frac{63}{2}$

b. Solve $s'(x) = 0$

We need to solve for the roots of $s'(x) = 0$:

$3x^2 - \frac{33}{2}x - \frac{63}{2} = 0$

This is a quadratic equation, and we can solve it using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For this quadratic equation, $a = 3$, $b = -\frac{33}{2}$, and $c = -\frac{63}{2}$. We can now substitute these values into the quadratic formula to find the solutions.

c. Study the sign of $s'(x)$ on the interval $[0; 10]$

Once we have the solutions from part b, we can use those critical points to test the sign of $s'(x)$ on the interval $[0; 10]$ by checking intervals determined by these points.

d. Complete the sign chart and variation table

In this part, we fill in the sign chart for $s'(x)$ and the variation table for $s(x)$. We will identify the behavior of $s(x)$ (increasing, decreasing, or constant) based on the sign of $s'(x)$.

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