Evgeny.Makarov said:If the values of both functions are nonnegative, then the product of minima is less than or equal to the minimum of products. This relies on the following fact about real numbers: if $0\le x_1\le y_1$ and $0\le x_2\le y_2$, then $x_1x_2\le y_1y_2$.
For example, consider functions $f$ and $g$ and just two values in the domain: $x_1$ and $x_2$. Let $a_1=f(x_1)$, $a_2=f(x_2)$, $b_1=g(x_1)$, $b_2=g(x_2)$, Then $\min(a_1,a_2)\le a_1$ and $\min(b_1,b_2)\le b_1$, so by the fact above $\min(a_1,a_2)\min(b_1,b_2)\le a_1b_1$. Similarly $\min(a_1,a_2)\min(b_1,b_2)\le a_2b_2$, so $\min(a_1,a_2)\min(b_1,b_2)\le\min(a_1b_1,a_2b_2)$.
If the numbers can be negative, then this conclusion no longer holds. For example, if $a_1=1$, $a_2=2$ and $b_1=b_2=-1$, then $\min(a_1,a_2)\min(b_1,b_2)=1\cdot(-1)=-1>-2=\min(-1,-2)=\min(a_1b_1,a_2b_2)$,
The minimum of product of 2 functions is the smallest possible value that can be obtained by multiplying two functions together. This value is typically found at the point where both functions are at their minimum values.
The minimum of product of 2 functions is calculated by finding the critical points of both functions and then evaluating the product of these critical points. The critical points are the points where the derivative of a function is equal to zero.
Yes, the minimum of product of 2 functions can be negative if one or both of the functions have negative values at their critical points. This means that the product of these values will also be negative, resulting in a negative minimum value.
The minimum of product of 2 functions is important in mathematics as it helps in finding the optimal solution to various problems. It is also used in optimization and calculus to find the minimum value of a function and to solve equations with multiple variables.
Yes, the minimum of product of 2 functions has many real-world applications, such as in economics, engineering, and physics. It is used to find the minimum cost, the most efficient design, and the optimal solution in various industries and fields.