need help asap test tomorrow 5. [10 points) Determine the absol ute maximum and

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need help asap test tomorrow

5. [10 points) Determine the absol ute maximum and the absolute minimum of (x) 22-9In r on the interval 4]. 6. [10 points] A spherical balloon is being filled at the rate of 2 em*/second. How fast is the radius increasing when the radius is 3 cm? The volume of a sphere with radius r is given by the formula V 7. [14 points] An island is 2 miles due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6 miles west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8 mph and swims at a rate of 3 mph. How far should the visitor run before swimming to minimize the time it takes to reach the island? 8. [18 points] Determine the domain and range of f(x), where it is increasing and decreasing, and where it is concave up and concave down. Find all the local extrema, intercepts, inflection points, and asymptotes. Sketch the graph of f (x). 12-21 +4 r(r - 4) (x - 2)2 f(x)- f"(x) = (z-2)"

Explanation / Answer

f = 2x^2 - 9lnx

Deriving :
f' = 4x - 9/x = 0

4x = 9/x

x^2 = 9/4

x = -3/2 and 3/2
Ob, we will eliminate -3/2 cuz it doesnt lie in [1/4 , 4]

So, consider x = 1/4 , 3/2 and 4

When x = 1/4, y = 12.6016 approx
When x = 3/2, we get y = 0.8508
When x = 4, we get y = 19.523

So, abs max = 19.523 when x= 4
Abs min = 0.8508 when x = 3/2

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6)
V = 4pi*r^3/3

Deriving with time,
dV/dt = dV/dr * dr/dt

dV/dt = 4pi*r^2 * dr/dt

Now, plug in :
2 = 4pi * 3^2 * dr/dt

2 = 36pi * dr/dt

dr/dt = 2/(36pi)

dr/dt = 1/(18pi) cm/second ----> ANS

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