Answer to Question #129847 in Calculus for Hetisani Sewela

Question #129847

Let f be the function

f (x) = x2 − ln x,

where x > 1.

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

falls. (5)

(b) Determine the coordinates of the local extreme point(s). (2)

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

down.


1
Expert's answer
2020-08-24T19:21:12-0400
"Solution"

a. ) Given

"f(x)\\ =\\ x^2-ln\\ x\\\\\nf'(x)\\ =\\ 2x-\\ \\frac{1}{x}\\\\\nf'(x)\\ >\\ 0\\ on\\ (1,\\infin)"


hence f rises on interval "(1,\\infin)" and f does not fall on x>1

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


"x^2-ln\\ x\\" has no critical points, therefore there are no local extreme points.


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

down.

"f(x)\\ =\\ x^2-ln\\ x\\\\\nf'(x)\\ =\\ 2x-\\ \\frac{1}{x},\\\\\nf''(x)\\ =\\ 2+\\ \\frac{1}{x^2}\\\\\nf''(x)\\ >\\ 0\\ on\\ (1,\\infin)"

hence f is concave up at x>1 on the graph



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