Final answer:
To meet the conditions for the function g, one could use a degree 3 polynomial function with careful selection of coefficients. By incorporating the specific values for g at -2 and 2, along with the derivative constraints, a system of equations can be formed to find an appropriate set of coefficients.
Step-by-step explanation:
To create a function g such that g(-2) = -1 and g(2) = 6, while also guaranteeing that the derivative g'(x) < 1 for all values of x, one can consider using a polynomial function of degree at least 3. By including constraints on the function's values and its derivative, we are in essence adding conditions to our polynomial coefficients.
Examples of such polynomial functions include cubic polynomials, which can be adjusted to fit the given points and ensure the required property on the derivative. Let's consider g(x) = ax^3 + bx^2 + cx + d. We would solve for a, b, c, and d using the given conditions and additional constraints derived from the derivative requirement.
For instance, a cubic equation where the coefficient of the x^3 term is small enough could satisfy g'(x) < 1. The fact that g(-2) = -1 and g(2) = 6 provides us with two equations. Two more equations would be needed to solve for all four coefficients a, b, c, and d. One way to get these extra equations could be by setting the derivative at the given points: if we want the slopes of the tangent lines at x = -2 and x = 2, we can state, for instance, g'(-2) = m and g'(2) = n, where m and n are any constants less than 1. This set of four equations would allow us to solve for the coefficients.