We need to find the minimum value of the function:

$E = a^2 + b^2 - 2ab - a - b - 2021$

Step 1: Rewrite the Expression

Rearrange the terms:

$E = (a^2 - 2ab + b^2) - a - b - 2021$

Recognizing that the first three terms form a perfect square:

$E = (a - b)^2 - a - b - 2021$

Step 2: Define a New Variable

Let $x = a - b$, then:

$E = x^2 - a - b - 2021$

Since $a = x + b$, substitute this into the equation:

$E = x^2 - (x + b) - b - 2021$

$E = x^2 - x - 2b - 2021$

Step 3: Minimize $E$

We observe that $x^2 - x$ is always non-negative for real numbers, so we aim to minimize $x^2 - x - 2b - 2021$.

To achieve the minimum, we choose $x = 0$ (since squares are non-negative, this gives the smallest value):

$E_{\min} = -2b - 2021$

Since $b$ can be arbitrarily large (negative), $E$ has no lower bound. Thus, there is no absolute minimum for all real numbers.

However, if we assume integer values for $a$ and $b$, we can minimize within constraints. Let me know if you'd like to solve in a specific domain!

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Related Questions:

1. What happens if $a$ and $b$ are restricted to positive integers?
2. How would the function change if a different coefficient replaced $-2ab$?
3. How do we find the critical points of multivariable quadratic functions?
4. What is the geometric interpretation of $E = (a-b)^2 - a - b - 2021$?
5. How can we use derivatives to confirm the minimum of a function?

Tip:

When dealing with quadratic expressions, completing the square often helps reveal minimum or maximum values more clearly.