Question

For Each Function Find Any Critical Values & Any Relative Extrema.

A g(x)=x3−12x+520.
B h(x)=x3−6x2−15x+1

Answer 1

To find critical values and relative extrema for the given functions, we need to first find the derivative of each function. The critical values for function g(x) are x = 2 and x = -2, and the critical values for function h(x) are x = 5 and x = -1.

Step-by-step explanation:

To find critical values and relative extrema for the given functions, we need to first find the derivative of each function. Let's start with function g(x):

g(x) = x^3 - 12x + 520

Taking the derivative of g(x), we get:

g'(x) = 3x^2 - 12

Setting g'(x) equal to zero and solving for x, we find the critical values:

3x^2 - 12 = 0

x^2 - 4 = 0

(x - 2)(x + 2) = 0

x = 2 or x = -2

Next, let's find the derivative of function h(x):

h(x) = x^3 - 6x^2 - 15x + 1

Taking the derivative of h(x), we get:

h'(x) = 3x^2 - 12x - 15

Setting h'(x) equal to zero and solving for x, we find the critical values:

3x^2 - 12x - 15 = 0

Using the quadratic formula, we get:

x = (-(-12) ± √((-12)^2 - 4(3)(-15))) / (2(3))

x = (12 ± √(144 + 180)) / 6

x = (12 ± √(324)) / 6

x = (12 ± 18) / 6

x = 30/6 or x = -6/6

x = 5 or x = -1

Therefore, the critical values for function g(x) are x = 2 and x = -2, and the critical values for function h(x) are x = 5 and x = -1.

Answer 2

The question involves finding critical values and relative extrema for cubic functions g(x) and h(x). The process includes calculating the first derivatives, setting them to zero, and solving for x. The context provided includes additional unrelated mathematical concepts.

Step-by-step explanation:

The question asks for finding critical values and relative extrema for the given functions g(x)=x^3−12x+520 and h(x)=x^3−6x^2−15x+1. To do this, one would find the first derivative of each function, set it equal to zero and solve for x to find critical points. It is not clear from the question if the functions are restricted to a certain domain or if the entire real line is considered. For the function f(x), with f(x) = 20 over the interval [0, 20], this constant function has no critical points and no relative extrema since its slope is zero everywhere within that domain.

The median-median approach, quadratic formula, and the concept of outliers mentioned in the question's context are steps typically used in data analysis or finding solutions to second-degree polynomials but are not directly applied to finding critical points or relative extrema of the provided cubic functions.