Question

Find the value of \(x\) that maximizes
\(f(x) = \log (-20x + 12\sqrt{x}).\)
If there is no maximum value, write "NONE".

Answer 1

HI!!

Answer 2

looks like a calc problem right? take the derivative and find the critical points

Answer 3

actually since the log is an increasing function, forget that part and maximize \[-20x+12\sqrt{x}\]

Answer 4

maximize \(-20u^2+12u=-20(u^2-\frac35u)=-20(u-\frac3{10})^2+\frac95\) subject to \(u\ge0\)

Answer 5

which obviously attains a maximum value of \(9/5\) (when \(x=3/10\)), so the maximum value of our expression is \(\log(9/5)=\log9-\log5\)

Answer 6

** when \(u=3/10\) rather