Taylor's Theorem
Taylor's theorem is a mathematical theorem that allows us to approximate a function using its derivatives at a single point. It provides a way to express a function as an infinite sum of terms involving the function's derivatives evaluated at a specific point.

Function f(x)
Let's consider the function f(x) = 40. This is a constant function, meaning that it does not depend on the value of x. In other words, f(x) is equal to 40 for all values of x.

Taylor's Theorem for a Constant Function
Taylor's theorem can still be applied to a constant function, although the result is not as interesting as for more complex functions. The theorem states that for a constant function, the nth derivative of the function is always zero for n > 0. This means that all the terms in the Taylor expansion of a constant function except the first term are zero.

Taylor Expansion of f(x) = 40
Since f(x) = 40 is a constant function, the Taylor expansion of f(x) will only involve the zeroth derivative of f(x), which is simply the value of f(x) itself. Therefore, the Taylor expansion of f(x) = 40 is:

f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...

Visual Representation
Here is a visual representation of the Taylor expansion of f(x) = 40:

f(x) = 40 + 0(x - a) + 0(x - a)^2/2! + 0(x - a)^3/3! + ...

Conclusion
In conclusion, the Taylor expansion of the constant function f(x) = 40 only involves the first term, which is the value of the function itself. This is because all the higher-order derivatives of a constant function are equal to zero. Therefore, the Taylor expansion simplifies to f(x) = 40, indicating that the function is equal to 40 for all values of x.