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**Unformatted text preview: **5.
For the function f(x) = 3x4 +8x3 -18x2 +5, find
a) the critical numbers;
J' ( x ) = 12X 3+24x = 36 X
fi( x ) = 0
12 x 3+ 24 X - 36 * = 0
12 x ( x 2+ 2 X - 3 )20
X = 0 , - 1, 3
b) the open intervals where the function is increasing;
Increasing ( 0, 20 )
c) the open intervals where the function is decreasing.
decreasing (-20, 01/ /
The total profit P(x) (in thousands of dollars) from the sale of x units of a certain prescription drug is
given by
P(x) = In(-x3 + 3x2 +72x+1) for x in [0, 10].
a) Find the total number of units that should be sold in order to maximize the total profit.
P' (X ) =
* 3 + 3 *2+ 72x +1
( - 3 x 2 + 6* + 72 ) 0
pi(x ) = 0 - -X3 + 3x 2 + 72x+1
( - 3 x 2 +48 + 72 ) = 0 -- 3 x 2 +4x + 72 = 0
- 3 (x + 4 ) ( x - 6 ) = 0 - X = - 4 6'
24 x = 0 -# In(1 ) = 0
* = 6 -D In ( 325 )= 5.78383
5 #
to
b) What is the maximum profit?
in thayanet )
= 10 - In ( 21 ) = 3. 04452/
(PCX ) = 5.78383 /
Given f(x) = -2x3 -9x2 +108x-10, find the open intervals where the function is concave upward
neave downward. Find any points of inflection.
+ to
Concave upward
(x ) = - 6x218 x+ 108
1- 3 00 )
< 2 - 18 x + 108 = 0
- 12x - 18 = 0
- 12x = 18
Concave downward
2 + 3x - 18) = 0
-12 - 12
X - 3 ) ( * +6 )
X = - 3
4 of 7
N/
M 5
POT =(- 3
- 371
f" ( - 5) =427
2
2...

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- Fall '19