What is the minimum value of the function f(x) = (x – 3)2 + (5 – x)2?

To find the minimum value of the function f(x) = (x – 3)2 + (5 – x)2, we can follow these steps:

1. Expand the squared terms: f(x) = (x – 3)(x – 3) + (5 – x)(5 – x)

2. Simplify the expression: f(x) = (x^2 – 6x + 9) + (25 – 10x + x^2) f(x) = x^2 – 6x + 9 + 25 – 10x + x^2

3. Combine like terms: f(x) = 2x^2 – 16x + 34

4. To find the minimum value, we can use the vertex formula: The x-coordinate of the vertex is given by x = -b/2a, where a = 2 and b = -16.

   x = -(-16) / (2 * 2) x = 16 / 4 x = 4

5. Substitute the x-coordinate of the vertex back into the original function to find the minimum value: f(4) = 2(4)^2 – 16(4) + 34 f(4) = 32 – 64 + 34 f(4) = 2

Therefore, the minimum value of the function f(x) = (x – 3)2 + (5 – x)2 is 2.