Given xradicalx+1 find the intervals on which f is increasing or decreasing, the local maximum and minimum values, the intervals of concavity and the inflect

Question

Given  xradicalx+1   find the intervals on which f is increasing or decreasing, the local
maximum and minimum values, the intervals of concavity and the inflection points. Sketch the graph.

 

 

xradicalx+1    Note: x+1 is under one radical

Answer 1

f(x) = x(x + 1)^1/2

 

 

First, we find the domain of f(x).

 

x + 1 ≥ 0

 

x ≥ -1

 

 

Domain is in the interval [-1, ∞).  This tells you that all critical points and turning points must happen within this interval.

 

Set the derivative of f(x) equal to zero to find extreme values.

 

(x + 1)^1/2 + (1/2)x(x + 1)^-1/2 = 0

 

 

Solve for x.

 

(x + 1)^-1/2[(x + 1) + (1/2)x] = 0

 

(x + 1)^-1/2((3/2)x + 1) = 0

 

 

Set the term in brackets equal to zero.

 

(3/2)x + 1 = 0

 

(3/2)x = -1

 

x = -2/3

 

 

Next, evaluate f'(-1) and f'(0).

 

If  f'(-1) is negative and f'(0) is positive, then f(-2/3) is a minimum.

If  f'(-1) is positive and f'(0) is negative, then f(-2/3) is a maximum.

 

 

Now take the second derivative of the expression and set it equal to zero.

 

(-1/2)(x + 1)^-3/2((3/2)x + 1) + (x + 1)^-1/2(3/2) = 0

 

(-1/2)(x + 1)^-3/2[(3/2)x + 1 - 3(x + 1)] = 0

 

 

Set the term in brackets equal to zero.

 

(-3/2)x - 2 = 0

 

(-3/2)x = 2

 

x = 2(-2/3)

 

x = -4/3

 

 

Since this value of x is not in the domain, the function has no infection points.