Consider the function y = x2 - 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 isa)1b)3c)4d)9Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given y = x² - 6x + 9

Or x = 3

Checking boundry conditions, we find that at x = 2, y = 1 and at x = 5, y = 4

Answer

Explanation:
To find the maximum value of the function y = x^2 - 6x + 9 over the interval [2, 5], we need to evaluate the function at the critical points and endpoints of the interval and compare the values.

Finding the Critical Points:
To find the critical points, we take the derivative of the function with respect to x and set it equal to zero.
Let's find the derivative of the function y = x^2 - 6x + 9:

dy/dx = 2x - 6

Setting dy/dx = 0, we have:

2x - 6 = 0
2x = 6
x = 3

So, x = 3 is the critical point of the function.

Evaluating the Function at Critical Points and Endpoints:
We need to evaluate the function y = x^2 - 6x + 9 at the critical point x = 3 and the endpoints x = 2 and x = 5.

When x = 3:
y = (3)^2 - 6(3) + 9
y = 9 - 18 + 9
y = 0

When x = 2:
y = (2)^2 - 6(2) + 9
y = 4 - 12 + 9
y = 1

When x = 5:
y = (5)^2 - 6(5) + 9
y = 25 - 30 + 9
y = 4

Comparing the Values:
Comparing the values of y at the critical point and endpoints, we find that y = 0 is the maximum value of y obtained when x varies over the interval [2, 5].

Therefore, the correct answer is option C) 4.

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