Thread: critical points and local extrema

1. critical points and local extrema

find the critical points and classify local extrema.

f(x)= 2-3x/2+x

2. Originally Posted by eayettey
find the critical points and classify local extrema.

f(x)= 2-3x/2+x

Is the function

$\displaystyle f(x)= 2-\frac{3x}{2}+x$

or

$\displaystyle f(x)= \frac{2-3x}{2+x}$

??

3. critical points and local extrema

4. Thanks for clearing that up.

If you divide the function as suggested then f(x) will be.

$\displaystyle \frac{2-3x}{2+x} = \frac{4}{2+x}-3$

so

$\displaystyle f(x) = \frac{4}{2+x}-3$

This suggests a vertical aymptote at x = -2 and a horizontal aymptote at y = -3.

To find a y-intercept make x = 0

$\displaystyle f(0) = \frac{4}{2+0}-3 = 2-3 = -1$

To find a x-intercept make y = 0

$\displaystyle 0 = \frac{4}{2+x}-3$

quotient rule

$\displaystyle 3(2+x) = 4$

$\displaystyle x = \frac{-2}{3}$

To find any maxima or minima you should use the
quotient rule , set your derivative equal to zero and solve for x.

quotient rule is, if

$\displaystyle y = \frac{u}{v}$

then

$\displaystyle y' = \frac{v\times u'-u\times v'}{v^2}$

I hope this helps some!

5. Originally Posted by eayettey
This is a rectangular hyperbola. This fact is made obvious in post #4. So it has no critical points.