Thread: Finding crritical points of a piecewise function

1. Finding crritical points of a piecewise function

Given:

$f(x)=\left\{\begin{array}{cc}-x^3+1 ,&\mbox{ if }
x\leq 0\\-x^2 +2x, & \mbox{ if } x>0\end{array}\right.$

Find the critical numbers and open intervals on which the function is increasing or decreasing.

Would I simply compute the derivative of -x^3+1 and -x^2+2x seperately and solve each down?

Can anyone offer some instruction for piecewise functions ?

2. Originally Posted by jimmyp
Given:

$f(x)=\left\{\begin{array}{cc}-x^3+1 ,&\mbox{ if }
x\leq 0\\-x^2 +2x, & \mbox{ if } x>0\end{array}\right.$

Find the critical numbers and open intervals on which the function is increasing or decreasing.

Would I simply compute the derivative of -x^3+1 and -x^2+2x seperately and solve each down?

Can anyone offer some instruction for piecewise functions ?
yes

all youre trying to do is find the same things you would in a regular function.

sketch the graph and then estimate where the critical values are. see if this is a good approximation of doing it analytically

3. Thus:

y' = $-3x^2$
and
$-2x+2$

$-3x^2=0$
$x=0$

and:

$-2x+2=0$
$x=1$

Critical point: $(0,1)$

And the test intervals are: $(- \infty, 0),(1, \infty)$

Thus:

$- \infty < x < 0$
test value: -2
Sign of f'(x) on -2x+2 = -2
Sign of f'(x) on -3x^2 = -12
Conclusion: decreasing for x < 0 and decreasing for x > 0

and:

$\infty < x < 1$
test value: 2
Sign of f'(x) on -2x+2 = -2
Sign of f'(x) on -3x^2 = -12
Conclusion: decreasing for x < 0 and decreasing for x > 0

is this correct or have i made a mistake ?

4. The intervals of increasing monotonicity are $(0,1)$ and the interval where $f(x)$ decreases are $(-\infty,0)\cup (1,\infty)$

You forgot to check for the sign of derivative on the interval (0,1)

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picewise functions critical points

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