Question

Determine determine the location of all critical points and determine the nature for the function f(x)=In(x^2+1)_x.​

Answer 1

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

Answer 2

Answer:

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Explanation:

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