Find the value of c guaranteed by the Intermediate Value Theorem. $$f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11$$
A) 0
B) 3
C) 5
D) 1
E) 4

Question

Quizplus · Accepted Answer

The Intermediate Value Theorem states that if a function is continuous on a closed interval, then it takes on every value between its minimum and maximum values on that interval. In this case, the function f(x) = x^2 - 2x + 8 is continuous on the interval [2,6]. The minimum value of f(x) on this interval is f(2) = 4, and the maximum value is f(6) = 20. Therefore, f(x) must take on every value between 4 and 20 on the interval [2,6]. Since f(c) = 11, c must be between 2 and 6. The only choice that satisfies this condition is B) 3.

Find the Value of C Guaranteed by the Intermediate Value f(x)=x2−2x+8,[2,6],f(c)=11f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11f(x)=x2−2x+8,[2,6],f(c)=11

Find the Value of C Guaranteed by the Intermediate Value $f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11$