The question asks to determine the maximum and minimum values of the function $f(x) = x^2 - 6x + 5$ within the interval $0 \leq x \leq 5$. There are three given statements to help solve the problem:

1. $f'(x) = 2x - 6$
2. $x = 3$
3. The maximum value is 5, and the minimum value is -4.

The task is to identify which combination of statements is sufficient to answer the question.

Step-by-step solution:

1. Derivative and Critical Points: The function's derivative is $f'(x) = 2x - 6$. Setting this equal to zero to find critical points:

   $2x - 6 = 0 \implies x = 3$

   This shows that $x = 3$ is a critical point.

2. Function Evaluation at Critical and Boundary Points: The function needs to be evaluated at the boundaries $x = 0$ and $x = 5$, and at the critical point $x = 3$.

   $f(0) = 0^2 - 6(0) + 5 = 5$ $f(5) = 5^2 - 6(5) + 5 = 25 - 30 + 5 = 0$ $f(3) = 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4$

   • Maximum value: $5$ at $x = 0$
   • Minimum value: $-4$ at $x = 3$

3. Analyzing the Statements:

   • Statement (1) provides the derivative, which helps find the critical point $x = 3$.
   • Statement (2) confirms the critical point $x = 3$.
   • Statement (3) gives the actual maximum and minimum values, which match the results of the function evaluations.

Conclusion:

• Statement (1) alone is enough to find the maximum and minimum values.
• Statements (2) and (3) by themselves are not sufficient without using (1).
• All three statements together are consistent and correct, but (1) alone is sufficient.

The correct answer is A.

Do you want further details or have any questions?

Related Questions:

1. How do you find critical points for a given function?
2. How is the first derivative used to determine maximum and minimum points?
3. What is the role of boundary points in optimization problems?
4. How do you evaluate the value of a function at critical and boundary points?
5. What is the significance of the second derivative in optimization?

Tip: Always check both the critical points and boundary points when looking for maximum and minimum values in a closed interval.