Give conditions on the constants b, c, and d so that the polynomial function f(x)=x^3+b x^2+c x+

step 1 for A and B that make this function F of X equals X cubed plus A, x plus B, always increasing. S

Give conditions on the constants b, c, and d so that the...

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Key Concepts

Monotonicity

Monotonicity refers to the property of a function consistently increasing or decreasing over its domain. A function is increasing if its derivative is nonnegative at every point of its domain, ensuring that the function never decreases.

Derivative of a Polynomial Function

The derivative of a polynomial is obtained by differentiating each term. For a cubic polynomial like f(x) = x³ + bx² + cx + d, the derivative is f'(x) = 3x² + 2bx + c. Analyzing the sign of this derivative helps determine whether the function is increasing over its entire domain.

Discriminant Analysis of Quadratic Functions

The expression 3x² + 2bx + c is a quadratic function in x. For a quadratic with a positive leading coefficient to be nonnegative for all x, its discriminant must be nonpositive. This means (2b)² - 4(3)(c) ? 0, which simplifies to b² - 3c ? 0. This condition is necessary to ensure that the derivative is nonnegative everywhere.

Effect of Constant Terms on Monotonicity

The constant term d in the polynomial f(x) = x³ + bx² + cx + d does not affect the derivative since its derivative is zero. Consequently, d has no impact on whether the function is increasing or not.

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