Who's up to the challenge? 4

Calculus Level 4

Given that $a,b$ and $c$ are positive real numbers, find the maximum value of

$F = \ln\left( a - \dfrac a{bc} \right) - bc - a .$

Round your answer to the nearest thousandth.

The answer is -3.58.

1 solution

Hamza A
Feb 6, 2016

$F=ln(a-\frac { a }{ bc } )-bc-a\\ \\ \nabla F=\begin{pmatrix} \frac { 1 }{ a } -1 \\ \frac { 1 }{ b(bc-1) } -c \\ \frac { 1 }{ c(bc-1) } -b \end{pmatrix}\\ \\ first\quad row\quad =0\Rightarrow a=1\quad \quad (1)\\ \\ \\ second\quad row\quad =0\Rightarrow \frac { 1 }{ b(bc-1) } =c\quad (2)\\ \\ third\quad row=0\Rightarrow \frac { 1 }{ c(bc-1) } =b\quad (3)\\ \\ \frac { 1 }{ bc-1 } =bc\\ \\ \Rightarrow bc=\phi \quad (dismising\quad (1-\sqrt { 5 } )/2\quad because\quad that\quad means\quad b\quad or\quad c\quad is\quad negative\quad \\ which\quad is\quad not\quad what\quad the\quad problem\quad is\quad asking\quad for)\\ \\ minimum\quad value=ln(1-\frac { 1 }{ \phi } )-\phi -1\approx -3.5805\\ \\ \\$

this is a maximum because the function is decreasing along the set of positive integers with a,b,c being positive .I took the Hessian to verify it but i didn't put it in the solution .I'll add it if someone is wishing to look at it

Sir, I got the solution set of $b$ and $c$ as $\pm1.272$ and $\pm0.786i$ . If we were to place the complex part in the function then it would yield $0.5804$ which is much larger than the actual solution $-3.580$ which is yielded by the real solution. Then why are we considering the solution yielded by real part?

Akshay Yadav - 5 years, 4 months ago

i didn't want to make it complicated,so i mentioned that a,b,c are positive real numbers,but your solution is completely valid if we were dealing with complex numbers

there are two places(possibly more because i didn't look for complex ones) when the function has a maximum(not necessarily because the gradient is zero)one is real and the other is imaginary,both are valid maximums(in their respective planes),but in the context of the problem,we shall only consider real valued variables,because they are simpler

i'll hopefully post a similar problem soon that deals with complex numbers,and if you wish to,you can provide a solution :)

Hamza A - 5 years, 3 months ago

Who's up to the challenge? 4

The answer is -3.58.

1 solution

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