quiquenobi2v6

2021-12-08

Find the absolute extreme values for $f\left(x\right)={\left(x+1\right)}^{2}+3$ on the interval [-2,2].

Carl Swisher

Step 1
To find : the absolute extreme values for $f\left(x\right)={\left(x+1\right)}^{2}+3$ on the interval [-2, 2].
Step 2
$f\left(x\right)={\left(x+1\right)}^{2}+3$
f′(x)=2(x+1)
To find: the critical values
Put f′(x)=0
$⇒2\left(x+1\right)=0$
$⇒2x+2=0$
$⇒2x=-2$
$⇒x=-1$
Step 3
To check f(x) at the endpoints and at the critical points.
$f\left(-2\right)={\left(\left(-2\right)+1\right)}^{2}+3=1+3=4$
$f\left(2\right)={\left(2+1\right)}^{2}+3=9+3=12$
$f\left(-1\right)={\left(\left(-1\right)+1\right)}^{2}+3=3$
Absolute maximum at x=2 and value=12
Absolute maximum at x=-2 and value=4

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