Question:medium

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

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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.
Updated On: Jan 25, 2026
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Solution and Explanation

The function is given by \( f(x) = x^2 - 2ax + b \). To find the values of \( a \) for which \( f(x) \) is increasing for \( x>0 \), we compute its first derivative and apply the condition for an increasing function. The first derivative is \( f'(x) = 2x - 2a \). For \( f(x) \) to be increasing for \( x>0 \), we require \( f'(x) \geq 0 \) for all \( x>0 \). This means \( 2x - 2a \geq 0 \) for \( x>0 \). Simplifying, we get \( x \geq a \) for \( x>0 \). This inequality holds for all \( x>0 \) if and only if \( a \leq 0 \). Therefore, the function \( f(x) \) is increasing for \( x>0 \) when \( a \leq 0 \). The set of values for \( a \) is \( \boxed{a \leq 0} \).
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