Determine determine the location of all critical points and determine the nature for the function f(x)=In(x^2+1)_x.
Answers
Answer:
Explanation:
f ’(x) indicates if the function is: Increasing or Decreasing on certain intervals.
Critical Point c is where f ’(c) = 0 (tangent line is horizontal), or f ’(c) = undefined (tangent line is vertical)
• f ’’(x) indicates if the function is concave up or down on certain intervals.
Inflection Point: where f '' ( x) = 0 or where the function changes concavity, no Min no Max.
If the sign of f ‘ (c) changes:
from + to - , then:
f (c) is a local Maximum
If the sign of f ‘ (c) changes:
from - to + , then:
f (c) is a local Minimum
If there is no sign change for f ‘ (c):
then f (c) is not a local extreme, it is:
An Inflection Point (concavity changes)
• Critical points, f ' (x) = 0 at : x = a , x = b
• Increasing, f ' (x) > 0 in : x < a and x > b
• Decreasing, f ' (x) < 0 in : a < x < b
• Max at: x = a , Max = f(a)
• Min at: x = b , Min = f(b)
• Inflection point, f '' (x) = 0 at : x = i
• Concave up, f '' (x) > 0 in: x > i
• Concave Down, f '' (x) < 0 in: x < i
• Critical points, f
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