MATH 1325 – EXAM 4 NAME: SHOW ALL WORK. ANSWERS WITHOUT WORK WILL RECEIVE NO CREDIT. YOU MUST USE A PENCIL. READ ALL DIRECTIONS. POINTS WILL BE DEDUCTED FOR FAILURE TO FOLLOW DIRECTIONS. TRUE/FALSE – WRITE THE WORD THAT BEST DESCRIBES THE GIVEN STATEMENT BY WRITING EITHER "TRUE" OR "FALSE" IN THE SPACE PROVIDED TO THE LEFT OF THE PROBLEM. 1. THE ABSOLUTE MAXIMUM OF A FUNCTION ALWAYS OCCURS WHERE THE DERIVATIVE HAS A CRITICAL FUNCTION. 2. IMPLICIT DIFFERENTIATION CAN BE USED TO FIND dy dx WHEN x IS DEFINED IN TERMS OF y . 3. IN A RELATED RATES PROBLEM, THERE CAN BE MORE THAN TWO QUANTITIES THAT VARY WITH TIME. 4. A CONTINUOUS FUNCTION ON AN OPEN INTERVAL DOES NOT HAVE AN ABSOLUTE MAXIMUM OR MINIMUM. 5. IN A RELATED RATES PROBLEM, ALL DERIVATIVES ARE WITH RESPECT TO TIME. MULTIPLE CHOICE – CHOOSE THE ONE ALTERNATIVE THAT BEST COMPLETES THE STATEMENT OR ANSWERS THE QUESTION BY CIRCLING THE CORRECT LETTER. 6. FIND THE MAXIMUM ABSOLUTE EXTREMUM AS WELL AS ALL VALUES OF x WHERE IT OCCURS ON THE SPECIFIED DOMAIN

Please see explanation column for answers. Also check number 6, its question is incomplete.  i used an assumed function, incase its not the same function with the one omitted, just follow steps

Step-by-step explanation: Questions 1-5 do not need any step by step explanation, its purely straight forward but Question 6 involves step by step explanation but  is not a complete question, though i used an assumed function.

FALSE   ---> 1. THE ABSOLUTE MAXIMUM OF A FUNCTION ALWAYS OCCURS WHERE THE DERIVATIVE HAS A CRITICAL FUNCTION. ___TRUE-->__ 2. IMPLICIT DIFFERENTIATION CAN BE USED TO FIND dy/dx WHEN x IS DEFINED IN TERMS OF y .

TRUE__--->3. IN A RELATED RATES PROBLEM, THERE CAN BE MORE THAN TWO QUANTITIES THAT VARY WITH TIME.

_FALSE  ---> 4. A CONTINUOUS FUNCTION ON AN OPEN INTERVAL DOES NOT HAVE AN ABSOLUTE MAXIMUM OR MINIMUM.

TRUE__---> 5. IN A RELATED RATES PROBLEM, ALL DERIVATIVES ARE WITH RESPECT TO TIME.

6. FIND THE MAXIMUM ABSOLUTE EXTREMUM AS WELL AS ALL VALUES OF x WHERE IT OCCURS ON THE SPECIFIED DOMAIN

Incomplete question Please.

But assuming the function f(x)= x³ -3x+1

 for (E)=(0,3)

step 1= let us use the power rule to find derivative of   f(x)= x^3 -3x+1

we will have f¹ (x) = 3x² -3

The critical values occurs when  3x² -3 = 0

which makes x=⁺₋1

As can be seen 3x² -3 = 0

                         3x²=3

                          x²=3/3=1

                       x= ⁺₋1

step 2=Now x= -1 cannot be considered because it is not in the interval  of the critical values (0,3)

therefore we consider x=1

step 3=The absolute extremes occurs at x=0, x=1, x=3 forf(x)= x³ -3x+1

when x=0,  f(0)= 0³-3(0)+ 1= 1

         x=1    f(1)=1³-3(1) +1=  -1

         x=3    f(3)= 3³ -3(3)+1= 19

Absolute minimum at x=1 has absolute value of-1

Absolute maximum of x=3 has absolute value of 19

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0

the discriminant is 64 + 4*1*12 = 112

this is prime.

step-by-step explanation: