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Question 1 of 5
Which of the following graphs is a negative definite?
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A Positive Definite is a function that is always positive for all values of xx. It is also a parabola that is concave up and is above the xx axis
A Negative Definite is a function that is always negative for all values of xx. It is also a parabola that is concave down and is below the xx axis
Check the characteristics of each graph to identify which is a Negative Definite
This parabola is concave up and is above the xx axis
Therefore, it is a Positive Definite
This parabola is concave down and is below the xx axis
Therefore, it is a Negative Definite
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Question 2 of 5
Identify whether the function below is a Negative Definite
y=-2x2+2x-1y=−2x2+2x−1
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A Negative Definite is a function that is always negative for all values of xx. It is also a parabola that is concave down and is below the xx axis
First, compute for the discriminant
ΔΔ |
== |
b2−4acb2−4ac |
Discriminant Formula |
|
== |
22−4(−2)(−1)22−4(−2)(−1) |
Substitute values |
|
== |
4-84−8 |
|
== |
-4−4 |
The value is negative, hence ΔΔ<<00
Next, check the value of aa
The value is negative, hence aa<<00
Therefore, we have established that the function is a Negative Definite
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Question 3 of 5
Identify whether the function below is a Positive Definite
2x2-x+3>02x2−x+3>0
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A Positive Definite is a function that is always positive for all values of xx. It is also a parabola that is concave up and is above the xx axis
First, compute for the discriminant
ΔΔ |
== |
b2−4acb2−4ac |
Discriminant Formula |
|
== |
−12−4(2)(3)−12−4(2)(3) |
Substitute values |
|
== |
1-241−24 |
|
== |
-23−23 |
The value is negative, hence ΔΔ<<00
Next, check the value of aa
The value is positive, hence aa>>00
Therefore, we have established that the function is a Positive Definite
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Question 4 of 5
For which values of kk will the function below be a Positive Definite
y=kx2+2kx+9y=kx2+2kx+9
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A Positive Definite is a function that is always positive for all values of xx. It is also a parabola that is concave up and is above the xx axis
First, compute for the discriminant
ΔΔ |
== |
b2−4acb2−4ac |
Discriminant Formula |
|
== |
2k2−4(k)(9)2k2−4(k)(9) |
Substitute values |
|
== |
4k2-36k4k2−36k |
Remember that for a function to be a Positive Definite, ΔΔ<<00
Substitute the ΔΔ computed previously, and then solve for kk
ΔΔ |
<< |
00 |
4k2-36k4k2−36k |
<< |
00 |
4k(k-9)4k(k−9) |
<< |
00 |
k=0k=0 |
|
k=9k=9 |
To determine which region around k=0k=0 and k=9k=9 would be included, plot these points and make a rough sketch of 4k2-36k4k2−36k
Replace the xx axis with kk axis and draw an upward parabola since 44 is positive
Remember that ΔΔ must be negative
Therefore, 00<<kk<<99
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Question 5 of 5
For which values of mm will the function below be a Positive Definite
mx2+4mx+16mx2+4mx+16
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A Positive Definite is a function that is always positive for all values of x. It is also a parabola that is concave up and is above the x axis
First, compute for the discriminant
Δ |
= |
b2−4ac |
Discriminant Formula |
|
= |
4m2−4(m)(16) |
Substitute values |
|
= |
16m2-64m |
Remember that for a function to be a Positive Definite, Δ<0
Substitute the Δ computed previously, and then solve for m
Δ |
< |
0 |
16m2-64m |
< |
0 |
16m(m-4) |
< |
0 |
m=0 |
|
m=4 |
To determine which region around m=0 and m=4 would be included, plot these points and make a rough sketch of 16m2-64m
Replace the x axis with m axis and draw an upward parabola since 16 is positive
Remember that Δ must be negative
Therefore, 0<m<4