Let g(x) =x/x^2+75 ..g'(x) = Ax2 + B(x2 + 75)C, where A = B =  C = g''(x) = Dx(x2 + E)(x2 + F)G, whereD = E = F = G =Find the interval(s) on which the graph of the function is concave upward and those on which it is concave downward. Enter the values for the endpoint(s) in the appropriate blanks and DNE in any empty blanks.The graph of the function is concave upward on the interval(s): (−∞, ∞)(−∞, a)    (−∞, a](a, ∞)[a, ∞)(−∞, a) ∪ (b, ∞)(−∞, a] ∪ [b, ∞)(−∞, a) ∪ (b, c)(a, b) ∪ (c, ∞)(a, b)[a, b]None of the above.a = b = c = The graph of the function is concave downward on the interval(s):(−∞, ∞)(−∞, d)    (−∞, d](d, ∞)[d, ∞)(−∞, d) ∪ (f, ∞)(−∞, d] ∪ [f, ∞)(−∞, d) ∪ (f, h)(d, f) ∪ (h, ∞)(d, f)[d, f]None of the above.d = f = h =

Question
Asked Oct 23, 2019
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Let g(x) =
x/x^2+75
 
..

g'(x) = 
Ax2 + B
(x2 + 75)C
, where
A = 
B =  
C = 


g''(x) = 
Dx(x2 + E)
(x2 + F)G
, where
D = 
E = 
F = 
G =



Find the interval(s) on which the graph of the function is concave upward and those on which it is concave downward. Enter the values for the endpoint(s) in the appropriate blanks and DNE in any empty blanks.


The graph of the function is concave upward on the interval(s):

 
(−∞, ∞)
(−∞, a)
    
(−∞, a]
(a, ∞)
[a, ∞)
(−∞, a) ∪ (b, ∞)
(−∞, a] ∪ [b, ∞)
(−∞, a) ∪ (b, c)
(a, b) ∪ (c, ∞)
(a, b)
[a, b]
None of the above.



a
b
c


The graph of the function is concave downward on the interval(s):

(−∞, ∞)
(−∞, d)
    
(−∞, d]
(d, ∞)
[d, ∞)
(−∞, d) ∪ (f, ∞)
(−∞, d] ∪ [f, ∞)
(−∞, d) ∪ (f, h)
(d, f) ∪ (h, ∞)
(d, f)
[d, f]
None of the above.



d
f
h

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Expert Answer

Step 1

The given function is

g(x)
x75
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g(x) x75

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Step 2

Obtain the first derivative as follows:

d
g'(x)
dx275
d
(x2+75)(x)(x2 +75)
(x2+75)
(x2 +75)(1)-x(2x)
dx
(x+75)
-x275
where A 1,B= 75,C
2
(x+75)
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d g'(x) dx275 d (x2+75)(x)(x2 +75) (x2+75) (x2 +75)(1)-x(2x) dx (x+75) -x275 where A 1,B= 75,C 2 (x+75)

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Step 3

Obtain the second deriva...

-x275
g'(x)
d
2 d
(x2 + 75) 75)-(-~2 +75)4( (x2 +75)
(62-75)
(-2x) (x+75)-4x (2 +75)(-x? +75)
- 75)- 4x(x2 +75)(-x? +75)
+75
2х (-х* + 225
+75
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-x275 g'(x) d 2 d (x2 + 75) 75)-(-~2 +75)4( (x2 +75) (62-75) (-2x) (x+75)-4x (2 +75)(-x? +75) - 75)- 4x(x2 +75)(-x? +75) +75 2х (-х* + 225 +75

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