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Mathematics · 2025 · 2 marks
CBSE 2025 · Region 1 · Set 2 · Q22
Find the values of 'a' for which f(x)=x^2-2 a x+b is an increasing function for x>0.
Find the values of 'a' for which $\displaystyle f(x)=x^{2}-2 a x+b$ is an increasing function for $\displaystyle x>0$.
Marking-scheme solution
\[\begin{aligned}
& f^{\prime}(x)=2 x-2 a \\
& 0<x<\infty \Rightarrow-2 a<2 x-2 a<\infty \Rightarrow-2 a<f^{\prime}(x)<\infty \\
& f(x) \text { is increasing iff } f^{\prime}(x) \geq 0 \\
& \Rightarrow-2 a \in[0, \infty) \Rightarrow a \in(-\infty, 0] \text { or }(-\infty, 0)
\end{aligned}
\]
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