zmudz
  • zmudz
Find the value of \(x\) that maximizes \(f(x) = \log (-20x + 12\sqrt{x}).\) Note: there may not be a maximum value at all.
Mathematics
katieb
  • katieb
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BAdhi
  • BAdhi
have you tried getting the derivative of f(x)?
anonymous
  • anonymous
Here's an approach that doesn't rely on calculus. Recall that \(\log z\) is defined for \(z>0\) (where \(z\in\mathbb{R}\)). In this case, \(z=12\sqrt x-20x\). You have \(z>0\) whenever \(12\sqrt x-20x>0\), or \(4\sqrt x(3-5\sqrt x)>0\). The factor \(4\sqrt x\) will always be positive, so you need to require that \(3-5\sqrt x>0\), i.e. \(x<\dfrac{9}{25}\). And because you're dealing with square roots, you're forced to work with \(x\) such that \(0

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