Find maxima/minima of polynomials

[Orson]

Homework Statement

find maxima/minima of following equation.

Homework Equations

-(x+1)(x-1)^2

The Attempt at a Solution

(-x-1)(x-1)^2

Using product rule, we obtain,
-1(x-1)^2+(-x-1)*2(x-1)
I don't know where to go from here. The software's factoring I had never seen before.

 

[LCKurtz]

Homework Statement

find maxima/minima of following equation.

Homework Equations

-(x+1)(x-1)^2

The Attempt at a Solution

(-x-1)(x-1)^2

Using product rule, we obtain,
-1(x-1)^2+(-x-1)*2(x-1)
I don't know where to go from here. The software's factoring I had never seen before.

Factor out an $(x-1)$ and simplify it. No software needed.

 

[Orson]

Factor out an $(x-1)$ and simplify it. No software needed.

How do i do that with the 2 and the -1?

and the (-x-1)

 

[LCKurtz]

Write $(x-1)(...)$, put what is left in the other parentheses, and simplify it.

 

[Orson]

Write $(x-1)(...)$, put what is left in the other parentheses, and simplify it.

-1(x-1)^2+(-x-1)*2(x-1)
=(x-1)((x-1)(-1)+2(-x-1))
ok. now what?

 

[LCKurtz]

-1(x-1)^2+(-x-1)*2(x-1)
=(x-1)((x-1)(-1)+2(-x-1))
ok. now what?

Simplify the second factor.

 

[Orson]

Simplify the second factor.

(x-1)(-3x-1)=0
x=1, -1/3

 

[LCKurtz]

(x-1)(-3x-1)=0
x=1, -1/3

OK, you have the critical values of $x$. Hopefully you can take it from here. It's sack time for me.

 

[Orson]

OK, you have the critical values of $x$. Hopefully you can take it from here. It's sack time for me.

I can. thank you very much. have a good night.

 

[Mark44]

Homework Statement

find maxima/minima of following equation.

Homework Equations

-(x+1)(x-1)^2

The Attempt at a Solution

(-x-1)(x-1)^2

Using product rule, we obtain,
-1(x-1)^2+(-x-1)*2(x-1)
I don't know where to go from here. The software's factoring I had never seen before.

Forget the software. A diff. rule you should have already seen is the constant multiple rule: $\frac d {dx}(k \cdot f(x)) = k \frac d {dx} f(x)$
Treat the minus sign as being a multiplier of -1, so $\frac d {dx}[-(x + 1)(x - 1)^2] = -1 \cdot \frac d{dx}[(x + 1)(x - 1)^2]$
If you can use the product rule on the part in the brackets, and bring along the factor of -1, you should be able to work through this.

 

[Ray Vickson]

Homework Statement

find maxima/minima of following equation.

Homework Equations

-(x+1)(x-1)^2

The Attempt at a Solution

(-x-1)(x-1)^2

Using product rule, we obtain,
-1(x-1)^2+(-x-1)*2(x-1)
I don't know where to go from here. The software's factoring I had never seen before.

Have you written the complete problem statement, with no clarifying words missing? I ask, because strictly speaking your function $p(x) = -(x+1)(x-1)^2$ does not have a maximum or a minimum when we place no restrictions on $x$. For any number $N > 0$, no matter how large, we can easily find values $x_1$ and $x_2$ giving $p(x_1) > N$ and $p(x_2) < -N$. (In other words, the maximum of $p(x)$ is $+\infty$ and its minimum is $-\infty$.) However, $p(x)$ does have local maxima and minima within restricted regions of $x$

 

[Orson]

Have you written the complete problem statement, with no clarifying words missing? I ask, because strictly speaking your function $p(x) = -(x+1)(x-1)^2$ does not have a maximum or a minimum when we place no restrictions on $x$. For any number $N > 0$, no matter how large, we can easily find values $x_1$ and $x_2$ giving $p(x_1) > N$ and $p(x_2) < -N$. (In other words, the maximum of $p(x)$ is $+\infty$ and its minimum is $-\infty$.) However, $p(x)$ does have local maxima and minima within restricted regions of $x$

Have you written the complete problem statement, with no clarifying words missing? I ask, because strictly speaking your function $p(x) = -(x+1)(x-1)^2$ does not have a maximum or a minimum when we place no restrictions on $x$. For any number $N > 0$, no matter how large, we can easily find values $x_1$ and $x_2$ giving $p(x_1) > N$ and $p(x_2) < -N$. (In other words, the maximum of $p(x)$ is $+\infty$ and its minimum is $-\infty$.) However, $p(x)$ does have local maxima and minima within restricted regions of $x$

It was multiple choice on khan academy. for which values of x is the function a local minimum or maximum. i can't remember which. But -1/3 was the correct answer per the software (s-word again)