Final answer:
The function g(x) = x⁴ + 4x³ has one relative maximum at x = -3 and no relative minimum.
Step-by-step explanation:
The function g(x) = x⁴ + 4x³ is a polynomial function. To find the relative extrema, we need to find the critical points of the function. The critical points occur where the derivative of the function is equal to zero or undefined.
Let's find the derivative of g(x) first:
g'(x) = 4x³ + 12x²
Setting g'(x) = 0, we can solve for the critical points:
4x³ + 12x² = 0
Factor out 4x²:
4x²(x + 3) = 0
Setting each factor equal to zero:
4x² = 0 → x = 0
x + 3 = 0 → x = -3
So there are two critical points: x = 0 and x = -3.
To determine the type of extrema at these critical points, we can use the second derivative test. Taking the second derivative:
g''(x) = 12x² + 24x
Substituting each critical point into g''(x), we can determine the concavity:
g''(0) = 0 → x = 0 is a point of inflection
g''(-3) = -36 → x = -3 is a relative maximum
Therefore, g(x) has one relative maximum at x = -3 and no relative minimum.