need some help on finding the critical value and x intercept of this equation:

y(x) = ax-xln(x)

thanksness

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- Mar 27th 2008, 12:44 PMmathleteFam. of func.
need some help on finding the critical value and x intercept of this equation:

y(x) = ax-xln(x)

thanksness - Mar 27th 2008, 01:11 PMteuthid
the x-intercept is the $\displaystyle x_0$ such that $\displaystyle y(x_0)=0$. so...

$\displaystyle 0=ax-x\ln x$

$\displaystyle \rightarrow ax=x\ln x$

$\displaystyle \rightarrow a = \ln x$

$\displaystyle \rightarrow x=e^{a}$

so the x-intercept of this function is $\displaystyle (e^{a},0)$.

What's the definition of a critical value? I'm not familiar with that term. - Mar 27th 2008, 04:56 PMtopsquark
A critical value would be a value where the first derivative of the function is 0 or where the function does not exist.

So the function does not exist at x = 0, so this is a critical point.

$\displaystyle y^{\prime}(x) = a - ln(x) - 1$

So when this is 0 we have a critical point:

$\displaystyle a - ln(x) - 1 = 0$

$\displaystyle ln(x) = a - 1$

$\displaystyle x = e^{a - 1}$

So this is your other critical point.

-Dan