Maximum and minimum of a function

In summary, the maximum and minimum of a function are the highest and lowest values that the function can achieve within a given domain. To find the maximum and minimum of a function, the derivative of the function can be taken and set equal to zero, and the resulting values of x will correspond to the maximum and minimum points of the function. There is a difference between local maximum/minimum and global maximum/minimum, with the former occurring in a specific region and the latter occurring in the entire graph. A function can have multiple maximum and minimum points, which can be used to optimize real-life situations, such as determining the most profitable price point or the most stable state of a system.
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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?
 
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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.
 

Related to Maximum and minimum of a function

1. What is the definition of a maximum and minimum of a function?

A maximum of a function is the highest value that the function reaches in a particular interval or over its entire domain. A minimum of a function is the lowest value that the function reaches in a particular interval or over its entire domain.

2. How can I find the maximum and minimum of a function?

The maximum and minimum of a function can be found by taking the derivative of the function and setting it equal to zero. The resulting values are the critical points of the function, which can then be plugged back into the original function to find the maximum and minimum values.

3. Can a function have more than one maximum or minimum?

Yes, a function can have multiple maximum and minimum points. These are known as local maximum and minimum points, and they occur when the slope of the function changes from positive to negative or vice versa.

4. How can I determine if a critical point is a maximum or minimum?

To determine if a critical point is a maximum or minimum, you can use the second derivative test. If the second derivative is positive at the critical point, then it is a minimum. If the second derivative is negative, then it is a maximum. If the second derivative is zero, then the test is inconclusive.

5. Is it possible for a function to have no maximum or minimum?

Yes, it is possible for a function to have no maximum or minimum. This occurs when the function is either constantly increasing or constantly decreasing over its entire domain. In this case, the function does not have any critical points.

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