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Consider the function y = x2 – 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is
- a)1
- b)4
- c)3
- d)9
Correct answer is option 'B'. Can you explain this answer?
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Consider the function y = x2 – 6x + 9. The maximum value of y ob...
Understanding the Function
The given function is y = x^2 – 6x + 9. This is a quadratic function, which can be analyzed for its maximum or minimum values.
Finding the Vertex
1. Standard Form
The function can be rewritten in vertex form. The general form of a quadratic function is y = ax^2 + bx + c. Here, a = 1, b = -6, and c = 9.
2. Vertex Calculation
The x-coordinate of the vertex can be found using the formula -b/(2a).
- b = -6
- a = 1
- x_vertex = -(-6)/(2*1) = 3
3. Finding y at Vertex
Substitute x = 3 back into the function:
y = (3)^2 - 6(3) + 9 = 9 - 18 + 9 = 0.
Evaluating the Interval
Next, we evaluate the function at the endpoints of the interval [2, 5].
1. At x = 2
y = (2)^2 - 6(2) + 9 = 4 - 12 + 9 = 1.
2. At x = 5
y = (5)^2 - 6(5) + 9 = 25 - 30 + 9 = 4.
Comparing Values
- At x = 2, y = 1.
- At x = 5, y = 4.
- At x = 3 (vertex), y = 0 (not within the interval).
Conclusion
The maximum value of y in the interval [2, 5] occurs at x = 5, where y = 4. Hence, the correct answer is option 'B'.
The given function is y = x^2 – 6x + 9. This is a quadratic function, which can be analyzed for its maximum or minimum values.
Finding the Vertex
1. Standard Form
The function can be rewritten in vertex form. The general form of a quadratic function is y = ax^2 + bx + c. Here, a = 1, b = -6, and c = 9.
2. Vertex Calculation
The x-coordinate of the vertex can be found using the formula -b/(2a).
- b = -6
- a = 1
- x_vertex = -(-6)/(2*1) = 3
3. Finding y at Vertex
Substitute x = 3 back into the function:
y = (3)^2 - 6(3) + 9 = 9 - 18 + 9 = 0.
Evaluating the Interval
Next, we evaluate the function at the endpoints of the interval [2, 5].
1. At x = 2
y = (2)^2 - 6(2) + 9 = 4 - 12 + 9 = 1.
2. At x = 5
y = (5)^2 - 6(5) + 9 = 25 - 30 + 9 = 4.
Comparing Values
- At x = 2, y = 1.
- At x = 5, y = 4.
- At x = 3 (vertex), y = 0 (not within the interval).
Conclusion
The maximum value of y in the interval [2, 5] occurs at x = 5, where y = 4. Hence, the correct answer is option 'B'.
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