find domain, range, max, min, inc, dec, con cavity, abs max min

Question

F(x)=2x⁴-3x³-10x²+20, find domain, range, maximum, minimum, increasing, decreasing, concavity, inflection points, absolute max/min?

Answer 1

Because F(x) is a polynomial, the domain and range is all real numbers.

 

To find the max and min, we set the derivative of F(x) equal to zero.  This is because the slope of the tangent line is the derivative, and the slope of the line tangent to F(x) max and min is has a slope of zero.

 

F'(x) = 0

 

8x³ - 9x² - 20x = 0

 

x(8x² - 9x - 20) = 0

 

 

Set the factors equal to zero.

 

x = 0            and            8x² - 9x - 20 = 0

 

 

Solve for x from the second equation.  Use FOIL.  If FOIL is not possible, then use the quadratic formula.

 

x = (9 ± √(81 - 4(-160))) / 16

 

x = (9 ± √(81 + 640)) / 16

 

x = (9 ± 26.851) / 16

 

x = (9 - 26.851) / 16                and                      x = (9 + 26.851) / 16

 

x = -1.1157                             and                      x = 2.2407

 

 

These x values are our critical points.  They are the location of the possible max and min.  Next, we use test points to evaluate the derivative of F'(x).  Use x=-2, x=-1, x=1, x=3.  If the derivative changes from negative to positive, then we have a minimum.  If the derivative changes from positive to negative, then we have a maximum.

 

F'(x) = 8x³ - 9x² - 20x

 

 

F'(-2) = -64 - 36 + 40

         = -60

 

F'(-1) = -8 - 9 + 20

         = 3

 

F'(1) = 8 - 9 - 20

        = -21

 

F'(3) = 216 - 81 - 60

        = 75

 

 

We have a minimum at F(-1.1157) and at  F(2.2407).

We have a maximum at F(0).

 

 

Intervals of increase are (-1.1157, 0)∪(2.2407, ∞).

Intervals of decrease are (-∞, -1.1157)∪(0, 2.2407).

 

 

 

To find the concavity and inflection points, we set the second derivative of F(x) equal to zero.

 

24x² - 18x - 20 = 0

 

2(8x² - 9x - 10) = 0

 

 

8x² - 9x - 10 = 0

 

 

x = (9 ± √(81 - 4(-80))) / 16

 

x = (9 ± 20.025) / 16

 

x = (9 - 20.025) / 16              and             x = (9 + 20.025) / 16

 

x = -0.689                             and             x = 1.814

 

 

These are our possible points of inflection.  We use test points to see where it is concave up and concave down.  If derivative changes signs, then we have point of inflection.  Evaluate the second derivative using test points.

 

F"(x) = 24x² - 18x - 20 

 

 

F"(-1) = 24 + 18 - 20

          = 22

 

F"(1) = 24 - 18 - 20

        = -14

 

F"(2) = 96 - 36 - 20

        = 40

 

F(x) is concave up in the interval (-∞, -0.689)∪(1.814, ∞).

F(x) is concave down in the interval (-0.689, 1.814).

 

Therefore, the points of inflection are F(-0.689) and F(1.814).