a.
Critical points occur where . The exponential factor is always positive, so we have
b. As the previous answer established, the critical point occurs at (-3, 8) if and .
c. Check the determinant of the Hessian matrix of :
The second-order derivatives are
so that the determinant of the Hessian is
The sign of the determinant is unchanged by the exponential term so we can ignore it. For and , the remaining factor in the determinant has a value of 4, which is positive. At this point we also have
which is negative, and this indicates that (-3, 8) is a local maximum.
The mean of sample means is the same as the mean of the underlying distribution.
The standard deviation of sample means is the original distribution sample mean divided by the square root of the number of samples.
Mean of sample mean distribution: 64
SD of sample mean distribution: 4/√64 = 1/2
Question 21
Let's complete the square
y = 3x^2 + 6x + 5
y-5 = 3x^2 + 6x
y - 5 = 3(x^2 + 2x)
y - 5 = 3(x^2 + 2x + 1 - 1)
y - 5 = 3(x^2+2x+1) - 3
y - 5 = 3(x+1)^2 - 3
y = 3(x+1)^2 - 3 + 5
y = 3(x+1)^2 + 2
Answer: Choice D
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Question 22
Through trial and error you should find that choice D is the answer
Basically you plug in each of the given answer choices and see which results in a true statement.
For instance, with choice A we have
y < -4(x+1)^2 - 3
-7 < -4(0+1)^2 - 3
-7 < -7
which is false, so we eliminate choice A
Choice D is the answer because
y < -4(x+1)^2 - 3
-9 < -4(-2+1)^2 - 3
-9 < -7
which is true since -9 is to the left of -7 on the number line.
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Question 25
Answer: Choice B
Explanation:
The quantity (x-4)^2 is always positive regardless of what you pick for x. This is because we are squaring the (x-4). Squaring a negative leads to a positive. Eg: (-4)^2 = 16
Adding on a positive to (x-4)^2 makes the result even more positive. Therefore (x-4)^2 + 1 > 0 is true for any real number x.
Visually this means all solutions of y > (x-4)^2 + 1 reside in quadrants 1 and 2, which are above the x axis.
Answer:
m∠C=28°, m∠A=62°, AC=34.1 units
Step-by-step explanation:
Given In ΔABC, m∠B = 90°, , and AB = 16 units. we have to find m∠A, m∠C, and AC.
As, cos(C)={15}/{17}
⇒ angle C=cos^{-1}(\frac{15}{17})=28.07^{\circ}\sim28^{\circ}
By angle sum property of triangle,
m∠A+m∠B+m∠C=180°
⇒ m∠A+90°+28°=180°
⇒ m∠A=62°
Now, we have to find the length of AC
sin 28^{\circ}=\frac{AB}{AC}
⇒ AC=\frac{16}{sin 28^{\circ}}=34.1units
The length of AC is 34.1 units