Find the critical values for X for the given polynomial by finding the X values
ID: 1167602 • Letter: F
Question
Find the critical values for X for the given polynomial by finding the X values where dy/dx = 0 and X where d2y/dx2 = 0. Tabulate these X values in increasing order, finding the Y value for each. Also find the sign (+ or -) of dy/dx and of d2y/dx2 at these values, as needed. Using the first and second derivative tests with the information you already calculated, determine which X value represents a maximum (MAX), which a minimum (MIN) and which an inflection point (INF). Label the points as such. Attach work to convince me you carried out these calculations.
Y = 3 –X2 –X3
X
-1
-.667
-.5
-.333
-.25
0
.25
Y
dy/dx
d2y/dx2
Label Point
(MAX, MIN, INF)
X
-1
-.667
-.5
-.333
-.25
0
.25
Y
dy/dx
d2y/dx2
Label Point
(MAX, MIN, INF)
Explanation / Answer
GIven the polynomial
Y=3 - x2 —x3
Dy/dx=-2x-3x2=0
=-x(2+3x)=0
Dy/dx at x=-1 will be
=-2*(-1) -3 (-1)2
=2-3=-1
Then either x=0 or x=-3/2, (simple quadratic solution)
Similarly
D2y /dx2= -2-6x=0 or x=-1/3 or -0.333
D2y /dx2 at x=-1
=-2-6*(-1)
=4
Now
Y=3 - x2 —x3
For x=-1
Y=3-(-1)2 - (-1)3
Y=3 -1 +1=3
Y=3 - x2 —x3
Y=3 – (-0.667)2 —(-0.667)3
Y=2.8518 Approx
Y=3 - x2 —x3
Y=3-(0.25)2-(0.25)3
Similarly all values will be calculated. The table has been filled up and values calculated usig spreadsheet.
X
-1
-0.667
-0.500
-0.333
-0.250
0.000
0.250
Y
3
2.852
2.766
2.926
2.953
3.000
2.922
DY/DX
-1
-0.001
0.250
0.333
0.313
0.000
-0.688
D2Y/DX2
4
2.002
1.000
-0.002
-0.500
-2.000
-3.500
Max or Min
Min
Min
Min
Max
Max
Max
Max
Constant
x^2
x^3
Result
Equation
dy/dx
0.250
0.250
-0.688
-2x-3x^2
y
3
0.250
0.250
2.922
Constant
x
Equation
d2y /dx2
-2
0.250
-3.500
-2-6x
The table above shows the calculation for x=0.25, similarly others were calculated.
X
-1
-0.667
-0.500
-0.333
-0.250
0.000
0.250
Y
3
2.852
2.766
2.926
2.953
3.000
2.922
DY/DX
-1
-0.001
0.250
0.333
0.313
0.000
-0.688
D2Y/DX2
4
2.002
1.000
-0.002
-0.500
-2.000
-3.500
Max or Min
Min
Min
Min
Max
Max
Max
Max
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.