# On what intervals the following equation is concave up, concave down and where it's inflection point is (x,y) #f(x)=x^8(ln(x))#?

##### 1 Answer

Nov 1, 2015

#### Answer:

- if
#0 < x < e^(-15/56)# then#f# is**concave down**; - if
#x > e^(-15/56)# then#f# is**concave up**; #x=e^(-15/56)# is a**(falling) inflection point**

#### Explanation:

To analyze concavity and inflection points of a twice differentiable function

- if
#f''(x_0)>0# , then#f# is*concave up*in a neighborhood of#x_0# ; - if
#f''(x_0)<0# , then#f# is*concave down*in a neighborhood of#x_0# ; - if
#f''(x_0)=0# and the sign of#f''# on a sufficiently small right-neighborhood of#x_0# is opposite to the sign of#f''# on a sufficiently small left-neighborhood of#x_0# , then#x=x_0# is called an*inflection point*of#f# .

In the specific case of

The first derivative is

The second derivative is

Let's study the positivity of

#x^6>0 iff x ne 0# #56ln(x)+15>0 iff ln(x)> -15/56 iff x>e^(-15/56)#

So, considering that the domain is

- if
#0 < x < e^(-15/56)# then#f''(x)<0# and#f# is**concave down**; - if
#x > e^(-15/56)# then#f''(x)>0# and#f# is**concave up**; - if
#x=e^(-15/56)# then#f''(x)=0# . Considering that on the left of this point#f''# is negative and on the right it is positive, we conclude that#x=e^(-15/56)# is a**(falling) inflection point**