Find a,b,c,d for Max $a+b-c+d$

• MHB
• anemone
In summary, the purpose of finding a, b, c, and d for Max $a+b-c+d$ is to determine the optimal values for each variable in order to maximize the expression $a+b-c+d$. This can be done using various methods such as algebraic manipulation, graphical analysis, or numerical techniques like calculus or optimization. Different variables can be used in place of a, b, c, and d as long as the constraints and relationships remain the same. Real-life applications of finding these values include economics, engineering, and physics. There is no specific formula or method for finding a, b, c, and d, but various mathematical and computational techniques can be used.
anemone
Gold Member
MHB
POTW Director
Let $a,\,b,\,c$ and $d$ be real numbers that satisfy the equation $f(x)=a\cos x+b\cos 2x+c\cos 3x+d\cos 4x \le 1$ for any real number $x$. Find the values of $a,\,b,\,c$ and $d$ such that $a+b-c+d$ takes the maximum number.

Since

$f(0)=a+b+c+d,\\f(\pi)=-a+b-c+d,\\f\left(\dfrac{\pi}{3}\right)=\dfrac{a}{2}-\dfrac{b}{2}-c-\dfrac{d}{2},$

then

$a+b-c+d=f(0)+\dfrac{2}{3}f(\pi)+\dfrac{4}{3}f\left(\dfrac{\pi}{3}\right)\le 3$ iff

$f(0)=f(\pi)=f\left(\dfrac{\pi}{3}\right)=1$, i.e. if $a=1,\,b+d=1$ and $c=-1$.

Let $t=\cos x,\,-1\le 4 \le 1$, then we have

\begin{align*}f(x)-1&=\cos x+b\cos 2x-\cos 3x+d\cos 4x-1\\&=t+(1-d)(2t^2-1)-(4t^3-3t)+d(8t^4-8t^2+1)-1\\&=2(1-t^2)[-4dt^2+2t+d-1)]\\&\le 0\end{align*}

That is, $4dt^2-2t+1-d\ge 0$

Taking $t=\dfrac{1}{2}+k,\,|k|<\dfrac{1}{2}$ we have

$k[2d-1+4dk]\ge 0$

We see that $d=\dfrac{1}{2}$, which gives

$4dt^2-2t+1-d=2t^2-2t+\dfrac{1}{2}=2\left(t-\dfrac{1}{2}\right)^2\ge 0$

So the maximum of $a+b-c+d$ is 3, where $(a,\,b,\,c,\,d)=\left(1,\,\dfrac{1}{2},\,-1,\,\dfrac{1}{2}\right)$.

1. What is the purpose of finding a,b,c,d for Max $a+b-c+d$?

The purpose of finding a,b,c,d for Max $a+b-c+d$ is to determine the maximum value that can be obtained by adding and subtracting the given variables. This can be useful in various mathematical and scientific applications, such as optimization problems and data analysis.

2. How do you find the values of a,b,c,d for Max $a+b-c+d$?

To find the values of a,b,c,d for Max $a+b-c+d$, you can use various methods such as algebraic manipulation, graphing, or numerical methods like calculus or linear programming. The specific method used will depend on the given constraints and the nature of the problem.

3. What are the possible constraints for finding a,b,c,d for Max $a+b-c+d$?

The constraints for finding a,b,c,d for Max $a+b-c+d$ can vary depending on the context of the problem. Some common constraints include a limited range of values for the variables, specific relationships between the variables, or limitations on the operations that can be performed.

4. Can the values of a,b,c,d for Max $a+b-c+d$ be negative?

Yes, the values of a,b,c,d for Max $a+b-c+d$ can be negative. This will depend on the given constraints and the specific problem being solved. In some cases, negative values may be necessary to achieve the maximum value, while in others they may not be allowed.

5. Are there any real-world applications for finding a,b,c,d for Max $a+b-c+d$?

Yes, there are many real-world applications for finding a,b,c,d for Max $a+b-c+d$. Some examples include optimizing production processes in manufacturing, maximizing profits in business, or determining the best investment strategy in finance. This concept can also be applied in various scientific fields, such as determining the optimal conditions for a chemical reaction or maximizing the efficiency of a mechanical system.

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