Question

Find the interval in which the function given by \[ f(x) = x^2 - 4x + 6 \] is decreasing.

Hint:

To find the interval of decrease, solve \( f'(x) < 0 \) to find where the slope of the function is negative.

Answer 1

Step 1: Find the first derivative of the function.
The first derivative of the function \( f(x) \) is: \[ f'(x) = \frac{d}{dx}(x^2 - 4x + 6) = 2x - 4 \]

Step 2: Find when the derivative is negative.
For the function to be decreasing, we need \( f'(x) < 0 \). So, solve: \[ 2x - 4 < 0 \] \[ 2x < 4 \] \[ x < 2 \]

Step 3: Conclusion.
The function \( f(x) = x^2 - 4x + 6 \) is decreasing for \( x < 2 \).