(a) Let f(x)=(1)/(3) x^3+x^2+k x, k a constant. For what values of k is f one-to-one? (b) Let g(

step 1 Hello. So here in part A, we have our function, F of X is equal to x cubed over 3 plus x squared

(a) Let f(x)=(1)/(3) x^3+x^2+k x, k a constant. For what...

Key Concepts

Injective Functions

Injective functions, also known as one-to-one functions, are functions where every element of the function's domain maps to a unique element in the codomain. This property is crucial for ensuring that the function has an inverse and that no two different inputs produce the same output.

Monotonicity and One-to-One Criteria

A common method for determining if a function is one-to-one is to check if it is monotonic—that is, either entirely non-decreasing or non-increasing. In calculus, if the derivative of a function does not change its sign over the entire domain, the function is monotonic and thus injective.

Using the Derivative to Test Monotonicity

The derivative of a function provides information about its rate of change. For functions that are polynomials, inspecting the sign of the derivative can help determine intervals over which the function is either strictly increasing or strictly decreasing. If the derivative maintains a consistent sign (or is zero only at isolated points such as at repeated roots), the function will be monotonic and thereby one-to-one.

Discriminant Analysis in Quadratic Functions

When the derivative of a function is a quadratic polynomial, its behavior (specifically whether it is always positive or always negative) can be studied by analyzing its discriminant—the term under the square root in the quadratic formula. A non-positive discriminant ensures that the quadratic does not have two distinct real roots, which in turn implies that the derivative does not change sign, confirming the monotonicity of the original function.

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