Urgency Can you help me Check these Applications of deritive.

Part A: For what intervals is the graph of h increasing? Decreasing? Justify your answer. (10 points)

In the interval [-10,-8], h' is positive (graph is above the x-axis), so h is increasing.
In the interval [-8,-2], h' is negative (graph is below the x-axis), so h is decreasing.
In the interval [-2,0], h' is positive (graph is above the x-axis), so h is increasing.
In the interval [0,6], h' is negative (graph is below the x-axis), so h is decreasing.


Part B: For what value(s) of x does h have relative extrema? Justify your answer. (5 points)

In the point (-8,0), the is increasing before and decreasing after, that is the point where h has a relative maximum exterma.

In the point (-2,0), the is decreasing before and increasing after, that is the point where h has a relative minimum exterma.

In the point (0,0), the is increasing before and decreasing after, that is the point where h has a relative maximum exterma.


Part C: For what intervals is the graph of h concave up? Concave down? Justify your answer. (10 points)

In the interval [-10,-6], h' is decreasing, so h"<0, so it is concave down.

In the interval [-6,-1], h' is increasing, so h">0, so it is concave up.

In the interval [-1,3], h' is decreasing, so h"<0, so it is concave down.

In the interval [3,6], h' is increasing, so h">0, so it is concave up.


Part D: For what value(s) of x does h have a point of inflection? Justify your answer. (5 points)

Where h" changes from positive to negative or vice-versa, it is the point of inflection. In point -6, -1, and 3, h" changes from positive to negative or vice-versa, so these are points of inflection.

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Question 2

Let's assume that the radius of the cone is r and the height is h. The cone is inscribed in the sphere so part of the height of the cone and the radius of the sphere form a right triangle. By using the Pythagorean theorem, we can write:

$r^{2} + x^{2} = 9^{2}$

And:

$h=9+x$

where:
x is from the center of sphere to the center of the cone.

To find the maximum volume, we need to express the volume of the cone in terms of a single variable. The volume of a cone is given by $V = \frac{1}{3} πr²h$ . We can rearrange the equation from step 1 to express h in terms of r:

$r^{2} = 9^{2} - x^{2}$

Substituting this value of h in the equation for volume, we get:

$V = \frac{1}{3} π(81-x^2)(9-x)$

To find the maximum volume, we take the derivative of V with respect to r and set it equal to zero, and solve for r. The resulting value of r will give us the radius of the cone of largest volume.

By solving the equation, we have:

$x=3$
$h=12 $
$r=\sqrt{72} $
$V=288\pi $

So, the exact dimensions of the cone are:

radius = 8.5 inches inches
height = 12 inches

And the volume is:

Volume = 904.8 cubic inches.


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