Considering this function, find the following properties?

Question

y=x^(1/5)*(x+6) 
a) Find the interval of increase. 

Find the interval of decrease. 

c) Find the inflection points. 

d) Find the intervals where the graph is concave upward. 

Find the interval where the graph is concave downward.

 

 

Answer 1

(a)

The Function is x^(1/5)(x+6)

Domain of the Function is (0, ∞)

F(x) = x^(1/5)(x+6)

Apply derivative on both sides.

F'(x) = x^(1/5) *1 + (x+6)(1/5) x^(-4/5)

F'(x) = x^(1/5) + (1/5) x^(-4/5) * x + 6 * (1/5) x^(-4/5)

F'(x) = x^(1/5) + (1/5) * x^(1/5) + (6/5)*x^(-4/5)

F'(x) = (6/5) * x^(1/5) + (6/5)*x^(-4/5)

Now f'(x) = 0

x^(1/5) + x^(-4/5) = 0

x^(1/5) +1/x^(4/5) = 0

(x+1)/x^(4/5) = 0

Then x + 1 = 0

x = -1

As -1 is not in the domain of f(x)

F(x) is increasing for any value of x over the interval of (0,∞).

Answer 2

(c)

The Function is x^(1/5)(x+6)

Domain of the Function is (0, ∞)

Apply First Derivative then F'(x) = (6/5) * x^(1/5) + (6/5)*x^(-4/5)

Apply derivative on both sides.

F''(x) = (6/5) x^(-4/5) + (6/5) x^(-9/5)

To find the inflection point F''(x) = 0

x+1 = 0

x = -1 which is not in the domain (0, ∞)

Therefore there are no inflection points of the function f(x) = x^(1/5)(x+6).

Answer 3

(d)

The Function is x^(1/5)(x+6)

Domain of the Function is (0, ∞)

There are no inflection points.

Interval  Test Value                 f''(x)                              Conclusion

(0, ∞)     x = 2        Concave Upwards.