Use calculus and properties of cubic polynomials to explain...

Use calculus and properties of cubic polynomials to explain why any polynomial function of the form $f(x)=$ $x^{4}+a x^{3}+b x^{2}+c x+d$ cannot be increasing on all of $(-\infty, \infty)$ or decreasing on all of $(-\infty, \infty)$

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots (considering multiplicities) in the complex numbers. For odd degree polynomials like a cubic derivative, there is always at least one real root. This real root is where the sign of the derivative changes, leading to the conclusion that the original function exhibits at least one turning point and therefore cannot be strictly monotonic over the entire domain.

Properties of Cubic Polynomials

When differentiating a quartic polynomial, the result is a cubic polynomial. Cubic polynomials, due to their odd degree and leading coefficient, tend to negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity. This guarantees that a cubic derivative must change sign at least once, which implies that the original quartic function cannot be monotonic over all real numbers.

Polynomial End Behavior

A polynomial’s end behavior describes how the function behaves as x tends to positive or negative infinity. For a monic even-degree polynomial, the function values approach infinity at both ends. This characteristic ensures that the function cannot maintain a single monotonic trend (strictly increasing or decreasing) over the entire real line because the function must change direction to connect its behavior at negative infinity with that at positive infinity.

Derivative and Monotonicity

The derivative of a function indicates its rate of change and thereby controls the function’s monotonicity. A function is increasing where its derivative is positive and decreasing where its derivative is negative. Hence, to have a function that is entirely increasing or entirely decreasing, its derivative must maintain a constant sign over all real numbers.

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