Show that f(x)=x^3+b x^2+c x+d is an increasing function if b^2<3 c. Find a condition on the coe

step 1 For this problem, we are asked first to show that f of x equals x cubed plus bx squared plus c i

Show that f(x)=x^3+b x^2+c x+d is an increasing function if...

Key Concepts

Derivative Test for Monotonicity

The derivative of a function represents its instantaneous rate of change. A function is strictly increasing if its derivative is positive for every value in its domain. This concept is a fundamental tool in calculus for determining the monotonic behavior of functions.

Polynomial Functions and Their Derivatives

Polynomial functions are expressed in terms of powers of x, and their derivatives are also polynomials. By differentiating a polynomial, we obtain another polynomial whose sign can provide insights into the original function's increasing or decreasing behavior over the entire real line.

Discriminant Analysis for Quadratic Polynomials

The discriminant of a quadratic polynomial, given by the expression under the square root in the quadratic formula, reveals the nature of the polynomial's roots. A negative discriminant, when coupled with a positive leading coefficient, ensures that the quadratic function does not have real roots and thus maintains a constant positive sign across all real numbers.

Global Positivity Conditions on Polynomials

When proving that a function is increasing, it is necessary to ensure that its derivative is positive for all values in the domain. For derivative expressions that are quadratic or reducible to a quadratic form, analyzing their global positivity often involves checking that no real roots exist or that the minimum value on a restricted domain is positive. This guarantees that the function retains a strictly increasing behavior.

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