Question

Find the values of \( a \) for which \( f(x) = x^2 - 2ax + b \) is an increasing function for \( x>0 \).

Hint:

To determine when a function is increasing, find its first derivative and set it greater than or equal to 0. Then solve for the variable and the condition on parameters.

Answer 1

The given function is: \[ f(x) = x^2 - 2ax + b. \] To determine for which values of \( a \) the function is increasing for \( x>0 \), we need to find the first derivative of \( f(x) \) and analyze the condition for increasing functions. The first derivative of \( f(x) \) is: \[ f'(x) = 2x - 2a. \] For the function to be increasing for \( x>0 \), we need \( f'(x) \geq 0 \) for all \( x>0 \). Thus, we need: \[ 2x - 2a \geq 0 \quad \text{for} \quad x>0. \] Simplifying: \[ x \geq a \quad \text{for} \quad x>0. \] For \( x>0 \), this inequality will hold true if \( a \leq 0 \). Thus, the values of \( a \) for which \( f(x) \) is increasing for \( x>0 \) are \( \boxed{a \leq 0} \).