Use the given function and the given interval to complete parts a and b. f (x )f(x)equals=negative 2 x cubed plus 21 x squared minus 60 x−2x3+21x2−60x on [1 comma 6 ][1,6] a. Determine the absolute extreme values of f on the given interval when they exist. b. Use a graphing utility to confirm your conclusions. a. What is/are the absolute maximum/maxima of f on the given interval? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The absolute maximum/maxima is/are nothingat xequals=nothing. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed.) B. There is no absolute maximum of f on the given interval.

(A)The absolute maximum of f(x) is at x=5

Step-by-step explanation:

F(x)=−2x³+21x²−60x on [1,6]

Find the first derivative.

f'(x)=-6x²+42x-60

Set the first derivative equal to zero.

-6x²+42x-60=0

Solve to find the critical points.

x=2,5

Next, we use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.

x=1,2,5,6

This is done by evaluating the function at x=1,2,5,6.

F(1)=−2(1)³+21(1)²−60(1)=-41

F(2)=−2(2)³+21(2)²−60(2)=-52

F(5)=−2(5)³+21(5)²−60(5)=-25

F(6)=−2(6)³+21(6)²−60(6)=-36

The maximum will occur at the highest f(x) value and the minimum will occur at the lowest f(x) value.

Absolute Maximum: (5,-25)

Absolute Minimum: (2,-52)

The graph is attached.

The absolute minimum and maximum of f(x) are (2,-52) and (5,-25) respectively.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

1. minimum

2. 6400

3. eight

hope this !