Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (g)/Solution 1

The given function is increasing where its first derivative is positive. Taking the derivative yields

$f^{\prime }(x)=2x-1/x$

We must now solve the inequality for x yields the interval where ƒ is increasing. We will only be interested in positive values of x, as negative values of x are not in the domain of the given function (because of the natural logarithm). Therefore,

$\begin{array}{rl}0& <2x-1/x,x>0\\ 1/x& <2x\\ 1& <2x^2\\ \frac{1}{2}& <x^2\\ \end{array}$

Therefore, we have

$x>+\frac{1}{\sqrt{2}}$

The negative solution was ruled out because of the requirement that x be a positive number.

The interval on which ƒ is increasing is

$(\frac{1}{\sqrt{2}},\mathrm{\infty })$