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
- if
0 < x < e^(-15/56) thenf is concave down; - if
x > e^(-15/56) thenf 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 , thenf is concave up in a neighborhood ofx_0 ; - if
f''(x_0)<0 , thenf is concave down in a neighborhood ofx_0 ; - if
f''(x_0)=0 and the sign off'' on a sufficiently small right-neighborhood ofx_0 is opposite to the sign off'' on a sufficiently small left-neighborhood ofx_0 , thenx=x_0 is called an inflection point off .
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) thenf''(x)<0 andf is concave down; - if
x > e^(-15/56) thenf''(x)>0 andf is concave up; - if
x=e^(-15/56) thenf''(x)=0 . Considering that on the left of this pointf'' is negative and on the right it is positive, we conclude thatx=e^(-15/56) is a (falling) inflection point