MHB Maximum and minimum of a function

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Let $a$ be an integer. Consider the function $y=\dfrac{12x^2-12ax}{x^2+36}$. For what integral values of $a$ the maximum and the minimum of the function $y=f(x)$ are integers?
 
Mathematics news on Phys.org
We have $(y-12)x^2+12ax+36y=0$.

This quadratic equation in $x$ has a solution in real numbers if and only if the discriminant $D=12^2a^2-4\cdot 36y(y-12)=12^2(a^2-y^2+12y)\ge 0$.

This happens if and only if $y^2-12y-a^2\le 0$ and this holds if and only if

$6-\sqrt{36+a^2}\le y\le 6+\sqrt{36+a^2}$

The end points of the above interval are the minimum and maximum of $y$. If we want them to be integers then $36+a^2=b^2$ must be a perfect square. This gives $36=(b+a)(b-a)$. If we take $36=mn$ as any factorization of 36 into positive integers, then $a=\dfrac{m-n}{2}$ and $b=\dfrac{m+n}{2}$. Now, $a,\,b$ are integers implies that $m,\,n$ are both odd or they are even. This gives two possibilities

$36=6\times 6$ or $36=2\times 18$

Now, $m=6=n$ gives $a=0$ and this forces $x^2=-\dfrac{36}{y-12}$ and this has a solution if and only if $0\le y \le 12$. As we are interested in the maximum and minimum $a\ne 0$ and we have the corresponding factorization $36=2\times 18$ which gives $a=\pm 8$ and $b=10$. So there are two integers $a=8$ and $a=-8$ with the required properties.
 
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Back
Top