Question

Determine whether the function \( f(x) = x^2 - 6x + 3 \) is increasing or decreasing in \( [4, 6] \).

Hint:

To determine whether a function is increasing or decreasing, find its derivative and check the sign of the derivative. If the derivative is positive, the function is increasing.

Answer 1

The given function is: \[ f(x) = x^2 - 6x + 3. \] Step 1: {Find the derivative of the function}
The derivative of \( f(x) \) is: \[ f'(x) = \frac{d}{dx}(x^2 - 6x + 3) = 2x - 6. \] Step 2: {Analyze the sign of the derivative in the interval \( [4, 6] \)}
For \( x \in [4, 6] \), we check the derivative: \[ f'(x) = 2x - 6. \] For \( x = 4 \), \( f'(4) = 2(4) - 6 = 8 - 6 = 2 \).
For \( x = 6 \), \( f'(6) = 2(6) - 6 = 12 - 6 = 6 \). Since \( f'(x)>0 \) for all \( x \in (4, 6) \), the function is increasing. Conclusion: The function is increasing over the interval \( [4, 6] \).