`y = x^2/2 - ln(x)` Locate any relative extrema and points of inflection.

Friday, October 29, 2010

$\displaystyle{y}=\frac{{{x}}^{{2}}}{{2}}-{\ln{{\left({x}\right)}}}$ Locate any relative extrema and points of inflection.

We are asked to locate the relative extema and any inflection points for the graph of $\displaystyle{y}=\frac{{{x}}^{{2}}}{{2}}-{\ln{{x}}}$ :

Extrema can occur only at critical points; i.e. points in the domain where the first derivative is zero or fails to exist. So we find the first derivative:

$\displaystyle{y}'={x}-\frac{{1}}{{x}}$ This is a continuous and differentiable function everywhere except x=0, which is not in the domain of the original function. (The domain, assuming real values, is x>0.)

Setting the first derivative equal to zero we get:

$\displaystyle{x}-\frac{{1}}{{x}}={0}=\Rightarrow{x}=\frac{{1}}{{x}}=\Rightarrow{{x}}^{{2}}={1}=\Rightarrow{x}={1}$ (x=-1 is not in the domain.)

For 0<x<1 the first derivative is negative, and for x>1 it is positive, so there is a minimum at x=1. This is the only minimum or maximum.

Inflection points can only occur when the second derivative is zero:

$\displaystyle{y}''={1}+\frac{{1}}{{{x}}^{{2}}}=\Rightarrow{y}''>{0}\forall{x}$ so there are no inflection points.

The graph:

Ninenotes

Friday, October 29, 2010

$\displaystyle{y}=\frac{{{x}}^{{2}}}{{2}}-{\ln{{\left({x}\right)}}}$ Locate any relative extrema and points of inflection.

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