function problem.?

Question

f(x) = 2x^3 - 54x^2 + 480x - 1300

a) what are the critical values?

b) do they represent either a relative minima/maxima?

c) find the coordinates of any points of inflection

d) evaluate the function (coordinates) at the critical points

im totally stuck. i have no clue how to do this. id really appreciate any help/comments on this. thanks!!

Answer 1

In mathematics, a critical point (or critical number) is a point on the domain of a function where...the derivative (or slope of the line when visualized) is equal to zero or a point where the function ceases to be differentiable....The value of a function at a critical point is called a critical value ("points" are inputs, "values" are outputs).

a) Set the derivative of the function with respect to x equal to zero, solve for all possible x values, and find f(x) at those values.

f'(x) = 6x^2 - 108x + 480 = 0
(-3x + 30)(-2x + 16) = 0
x = 10, 8
f(10) = 2000-5400+4800-1300 = 100
f(8) = 1024-3456+3840-1300 = 108

b) A critical value is a maxima if the second derivative at that point is negative because that means the slope is changing in a negative direction. Conversely, a minima is where the second derivative is positive.

f''(x) = 12x - 108
f''(10)=120-108 > 0. Therefore, f(10) is a local minima.
f''(8)=96-108 < 0. Therefore, f(8) is a local maxima.

c) No points of inflection. A point of inflection is a point at which the first and second derivatives are zero.

d) f(10) = 100 and f(8) = 108 (See answer to (a) above)

Note: Ironduke's answer below is helpful in that it also gives the roots of the function, but those are not the "critical values" of the function. As explained above, f(10) is a local minima, not an inflection point.

Answer 2

f(x) = 2x^3 - 54x^2 + 480x - 1300
f(x) = (x-5)(2x^2 -44x+260)
The function has one real root x=5 and two imaginary roots.
The function has a local max of 108 at x=8.
There is a point of inflection at x = 10, which is also a point of inflection.

Answer 3

g(a million)=-a million g(-a million)=a million a is 'no longer real' b)g(2)=8 g(-2)=4 g(2)+g(-2)=12 no longer real c)g(-2)=4 a million/2 g(2)= a million/2(8) =4 real d)g(0)=6 g(a million)=-a million g(-a million)=a million 6=a million+-a million fake e) g(g(-a million)) = g(a million)=-a million g(g(a million))=g(-a million)=a million fake The functionality has a pair. area and variety (-2,4) potential g(-2)=4 -2 in the area and four in the variety. (2,8) potential g(2)=8