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Question 1 of 2
Find the variables a,b,ca,b,c in f(x)=x3+f(x)=x3+aax2+x2+bbx+x+cc,
given that:
Stationary Point x=3x=3
Inflection Point (1,4)(1,4)
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A point that satisfies f′(x)=0 is called a Stationary Point
A point that satisfies f”(x)=0 is called an Inflection Point
First, equate f”(x) to 0 and substitute x=1 (from inflection point (1,4))
f(x) |
= |
x3+ax2+bx+c |
f′(x) |
= |
3x2+2ax+b |
f”(x) |
= |
6x+2a=0 |
Equate f”(x) to 0 |
f”(1) |
= |
6(1)+2a=0 |
Substitute x=1 |
6+2a |
= |
0 |
6+2a-6 |
= |
0-6 |
Subtract 6 from both sides |
2a |
= |
-6 |
2a÷2 |
= |
-6÷2 |
Divide both sides by 2 |
a |
= |
-3 |
Next, equate f′(x) to 0 and substitute the stationary point x=3 and a=-3
f(x) |
= |
x3+ax2+bx+c |
f′(x) |
= |
3x2+2ax+b=0 |
Equate f′(x) to 0 |
f′(3) |
= |
3(3)2+2(-3)(3)+b=0 |
Substitute x=3 and a=-3 |
3(9)+2(-9)+b |
= |
0 |
27-18+b |
= |
0 |
9+b |
= |
0 |
9+b-9 |
= |
0-9 |
Subtract 9 from both sides |
b |
= |
-9 |
Finally, substitute known values into the function to solve for c
a=-3
b=-9
Inflection Point (x,f(x))=(1,4)
f(x) |
= |
x3+ax2+bx+c |
4 |
= |
(1)3+(-3)(12)+(-9)(1)+c |
Substitute known values |
4 |
= |
1+(-3)(1)-9+c |
4 |
= |
1-3-9+c |
4 |
= |
-11+c |
4+11 |
= |
-11+c+11 |
Add 11 to both sides |
15 |
= |
c |
c |
= |
15 |
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Question 2 of 2
Find the variables a,b,c in f(x)=ax3+bx2+cx-5,
given that:
Stationary Point x=0
Inflection Point (2,-13)
Write fractions in the format “a/b”
Incorrect
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A point that satisfies f′(x)=0 is called a Stationary Point
A point that satisfies f”(x)=0 is called an Inflection Point
First, equate f′(x) to 0 and substitute x=0 (stationary point)
f(x) |
= |
ax3+bx2+cx-5 |
f′(x) |
= |
3ax2+2bx+c=0 |
Equate to 0 |
f′(0) |
= |
3a(0)2+2b(0)+c=0 |
Substitute x=0 |
Next, equate f”(x) to 0 and substitute x=2 (from inflection point (2,-13))
f(x) |
= |
ax3+bx2+cx-5 |
f′(x) |
= |
3ax2+2bx+c |
f”(x) |
= |
6ax+2b=0 |
Equate to 0 |
f”(2) |
= |
6a(2)+2b=0 |
Substitute x=2 |
|
|
12a+2b=0 |
Form another equation by substituting known values into the function
Inflection Point (x,f(x))=(2,-13)
c=0
f(x) |
= |
ax3+bx2+cx-5 |
-13 |
= |
a(23)+b(22)+(0)(2)-5 |
Substitute known values |
-13 |
= |
8a+4b+0-5 |
-13+13 |
= |
8a+4b+0-5+13 |
Add 13 to both sides |
0 |
= |
8a+4b+8 |
0÷2 |
= |
8a+4b+8÷2 |
Divide both sides by 2 |
0 |
= |
4a+2b+4 |
4a+2b+4 |
= |
0 |
Use elimination method on the two equations to solve for a
12a+2b=0–4a+2b+4=08a+0–4=0
8a-4 |
= |
0 |
8a |
= |
4 |
8a÷8 |
= |
4÷8 |
Divide both sides by 8 |
|
a |
= |
12 |
Substitute the value of a to one of the equations and solve for b
12a+2b |
= |
0 |
|
12(12)+2b |
= |
0 |
Substitute a=12 |
|
6+2b |
= |
0 |
2b |
= |
-6 |
2b÷2 |
= |
-6÷2 |
Divide both sides by 2 |
b |
= |
-3 |