Final answer:
To find the value of c such that f(x) = x^2 - 4x + 1 on the interval [0,2] is equal to f(c), you need to solve for c. The possible values of c are x and 4 - x.
Step-by-step explanation:
To find the value of c such that f(x) = x2 - 4x + 1 on the interval [0,2] is equal to f(c), we need to solve for c.
First, substitute f(x) into the equation: f(c) = c2 - 4c + 1.
Next, set f(c) equal to f(x): x2 - 4x + 1 = c2 - 4c + 1.
Simplify the equation: x2 - c2 - 4x + 4c = 0.
Now, factor the equation: (x - c)(x + c) - 4(x - c) = 0.
From the factored equation, we can see that x - c = 0 or x + c - 4 = 0.
Solve for c in each equation:
If x - c = 0, then c = x.
If x + c - 4 = 0, then c = 4 - x.
Therefore, the possible values of c are x and 4 - x.