If the function f (x) = x2– 8x + 12 satisfies the condition of Rolle’s Theorem on (2, 6), find the value of c such that f ‘(c) = 0a)6b)4c)8d)2Correct answer is option 'B'. Can you explain this answer?

Question

If the function f (x) = x2&#8211; 8x + 12 satisfies the condition of Rolle&#8217;s Theorem on (2, 6), find the value of c such that f &#8216;(c) = 0&#8203;a)6b)4c)8d)2Correct answer is option 'B'. Can you explain this answer?

Accepted Answer

f (x) = x² - 8x + 12
Function satisfies the condition of Rolle's theorem for (2,6).
We need to find c for which f’(c) = 0
f’(x) = 2x – 8
f’(c) = 2c – 8 = 0
c = 4

Answer

Just differentiate the eqn and take it equal to 0 u will get the value of x which will be the value of c

Answer

To apply Rolle's Theorem, we need to check three conditions:
1. The function must be continuous on the closed interval [a, b].
2. The function must be differentiable on the open interval (a, b).
3. The function must have the same values at the endpoints.

Let's check these conditions for the given function f(x) = x^2 - 8x + 12 on the interval (2, 6):

1. Continuity:
The function f(x) = x^2 - 8x + 12 is a polynomial function, and polynomial functions are continuous for all real numbers. Therefore, f(x) is continuous on the interval (2, 6).

2. Differentiability:
The function f(x) = x^2 - 8x + 12 is a polynomial function, and polynomial functions are differentiable for all real numbers. Therefore, f(x) is differentiable on the interval (2, 6).

3. Same values at endpoints:
We need to check if f(2) = f(6). Evaluating the function at these points:
f(2) = 2^2 - 8(2) + 12 = 4 - 16 + 12 = 0
f(6) = 6^2 - 8(6) + 12 = 36 - 48 + 12 = 0

Since f(2) = f(6) = 0, the function satisfies the third condition.

Since all three conditions are satisfied, we can apply Rolle's Theorem.

Rolle's Theorem states that if a function satisfies the above conditions, then there exists at least one value c in the open interval (a, b) such that f'(c) = 0.

In this case, f'(x) = 2x - 8. To find the value of c, we need to solve the equation f'(c) = 0:

2c - 8 = 0
2c = 8
c = 4

Therefore, the value of c such that f(c) = 0 is c = 4.

Hence, the correct answer is option B) 4.

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