Question

To find the domain and critical numbers/points of the function f(x)=x² √4−x​, you need to consider two aspects: the domain and the critical points, which occur where the derivative is zero or does not exist.

Answer 1

The domain of the function
`\( f(x) = x^2 √(4-x) \)` is
`\( x \leq 4 \)`. The critical points occur at ( x = 0 ) and ( x = 4 ).

Step-by-step explanation:

In the given function
`\( f(x) = x^2 √(4-x) \)`, we need to consider two aspects: the domain and the critical points. For the domain, the expression under the square root must be non-negative, so
`\( 4 - x \geq 0 \)`. Solving this inequality, we find
`\( x \leq 4 \)`, which means the domain is
`\( x \leq 4 \)`.

Next, to find the critical points, we need to calculate the derivative of
`\( f(x) \)` and set it equal to zero. The derivative f'(x) is found using the product rule and chain rule. After finding f'(x), we set it equal to zero and solve for ( x ). The critical points are the solutions to f'(x) = 0 or where f'(x) does not exist. In this case, we find critical points at ( x = 0 ) and ( x = 4 ).

To summarize, the domain of the function is
`\( x \leq 4 \)`, and the critical points occur at ( x = 0 ) and ( x = 4 ). These critical points are where the derivative is zero or does not exist, indicating potential points of interest in the behavior of the function.