Answer to Question #130384 in Calculus for sire

Question

Let f be the function f (x) = x^2 − lnx^8 where x > 1

(a) Use the sign pattern for f '(x) to determine the intervals where f rises and where f

falls

(b) Determine the coordinates of the local extreme point(s).

(c) Find f ''(x) and determine where the graph of f is concave up and where it is concave

down.

Answer 1

f(x)=x²-ln(x),

Becuse of the domain of the logarithmic function, one considers x>0.

f '(x)=2x-8/x,

f '(x)=0 x> 1

2x-8/x=0,

x-4/x=0,

x²-4=0, x> 1

x=2

f(2)=2*2-8(ln(2)=4-8*0.693=-1.545

x²-4=0, x> 1

x=2

If x> 2 the function f(x) increases.

If 0<x<2 the function f(x) decreases.

 f '' (x)=2+8/x²

 f '' (x)> 0

The function f(x) is concave upward.