Minimum

[Guest]

Hello, could You help me please find the minimum and the maximum of these functions on an interval:

1. y= x^3-3x+2, , x [-2;2]
2. y=-x^4+10x^2-9 , x [-3;3]
3. y= 3x^3-9x+3, x [4;6]
4. y= x^3-3x+3, x [-1;1] and x [1;3]

I would be very helpful.

[Baltuilhe]

Good morning!

1)
[tex]y=x^3-3x+2\\
y'=3x^2-3\\
y'=0\\
3x^2-3=0\\
3x^2=3\\
x^2=1\\
x=\pm 1\\
y''=6x\\
x=1\Rightarrow y''=6\therefore\text{MINIMUM}\Rightarrow y=0\\
x=-1\Rightarrow y''=-6\therefore\text{MAXIMUM}\Rightarrow y=4\\
x=-2\Rightarrow y=0\therefore\text{MINIMUM}\\
x=2\Rightarrow y=4\therefore\text{MAXIMUM}[/tex]

2)
[tex]y=-x^4+10x^2-9\\
y'=-4x^3+20x\\
y'=0\\
-4x^3+20x=0\\
-4x(x^2-5)=0\\
x=0\\
x^2-5=0\\
x^2=5\\
x=\pm\sqrt{5}\\
y''=-12x^2+20\\
x=0\Rightarrow y''=20\therefore\text{MINIMUM}\Rightarrow y=-9\\
x=-\sqrt{5}\Rightarrow y''=-40\therefore\text{MAXIMUM}\Rightarrow y=66\therefore\text{RELATIVE}\\
x=\sqrt{5}\Rightarrow y''=-40\therefore\text{MAXIMUM}\Rightarrow y=66\therefore\text{RELATIVE}\\
x=-3\Rightarrow y=162\text{MAXIMUM}\therefore\text{ABSOLUTE}\\
x=3\Rightarrow y=162\text{MAXIMUM}\therefore\text{ABSOLUTE}[/tex]

3)
[tex]y=3x^3-9x+3\\
y'=9x^2-9\\
y'=0\\
9x^2-9=0\\
9x^2=9\\
x^2=1\\
x=\pm 1\\
y''=18x\\
x=-1\Rightarrow y''=-18\therefore\text{MAXIMUM}\Rightarrow y=9\therefore\text{OUT RANGE}\\
x=1\Rightarrow y''=18\therefore\text{MINIMUM}\Rightarrow y=-3\therefore\text{OUT RANGE}\\
x=4\Rightarrow y=159\therefore\text{MINIMUM}\\
x=6\Rightarrow y=597\therefore\text{MAXIMUM}[/tex]

4)
[tex]y=x^3-3x+3\\
y'=3x^2-3\\
y'=0\\
3x^2-3=0\\
3x^2=3\\
x^2=1\\
x=\pm 1[/tex]
Between [-1;1]:
[tex]y''=6x\\
x=-1\Rightarrow y''=-6\therefore\text{MAXIMUM}\\
x=1\Rightarrow y''=6\therefore\text{MINIMUM}[/tex]

Between [1;3]:
[tex]x=1\Rightarrow y=1\therefore\text{MINIMUM}\\
x=3\Rightarrow y=21\therefore\text{MAXIMUM}[/tex]

[Guest]

Thank You!

[HallsofIvy]

Where did you get these problems? They are fairly standard introductory Calculus problem but you seem to be saying that you have never taken a Calculus problem!

[HallsofIvy]

Where did you get these problems? They are fairly standard introductory Calculus problem but you seem to be saying that you have never taken a Calculus problem!