Questions: Find the critical points of f. Assume a is a c f(x) = x sqrt(x-a)

Solution Steps

To find the critical points of the function \( f(x) = x \sqrt{x - a} \), we need to follow these steps:

1. Compute the first derivative of the function \( f(x) \).
2. Set the first derivative equal to zero and solve for \( x \) to find the critical points.
3. Check the domain of the function to ensure the critical points are valid.

The function is given by

\[ f(x) = x \sqrt{x - a} \]

To find the critical points, we first compute the first derivative \( f'(x) \):

\[ f'(x) = \frac{x}{2\sqrt{x - a}} + \sqrt{x - a} \]

Next, we set the first derivative equal to zero to find the critical points:

\[ \frac{x}{2\sqrt{x - a}} + \sqrt{x - a} = 0 \]

This simplifies to:

\[ \frac{x + 2(x - a)}{2\sqrt{x - a}} = 0 \]

Thus, we solve for \( x \):

\[ x + 2(x - a) = 0 \implies 3x - 2a = 0 \implies x = \frac{2a}{3} \]

The critical point \( x = \frac{2a}{3} \) must be checked against the domain of the original function. The function \( f(x) \) is defined for \( x \geq a \). Therefore, we need:

\[ \frac{2a}{3} \geq a \implies 2a \geq 3a \implies a \leq 0 \]

This means that the critical point is valid only if \( a \leq 0 \).

Final Answer