Let's break this task into smaller parts and work through each step systematically to solve the problem. The given function is:

$f(x) = \sqrt{4 \cdot x} \cdot \left( \frac{1}{5}x^2 - 3x \right)$

Task 1: Determine the y-intercept ($y_{AB}$) and the roots ($x_N$):

• The y-intercept occurs when $x = 0$. Substitute $x = 0$ into $f(x)$.
• To find the roots ($x_N$), solve $f(x) = 0$.

Task 2: Calculate the derivatives:

1. First derivative $f'(x)$
2. Second derivative $f''(x)$
3. Third derivative $f'''(x)$

Task 3: Tangent at $x_S = 4$:

The tangent equation at $x_S = 4$ is given by: $y_T(x) = f'(x_S) \cdot (x - x_S) + f(x_S)$ Substitute $x_S = 4$ into $f(x)$ and $f'(x)$.

Task 4: Determine extrema and inflection points:

1. Find critical points where $f'(x) = 0$.
2. Determine the type of critical points (minima or maxima) using $f''(x)$.
3. Find inflection points ($x_W$) where $f''(x) = 0$ and confirm changes in concavity using $f'''(x)$.

Would you like me to calculate and provide detailed solutions for each step? If yes, I'll begin with Task 1. Let me know!