MHB Show all real roots are negative

[anemone]

Show that all real roots of the polynomial $f(x)=x^5-10x+38$ are negative.

Note:

I know this is a fairly easy challenge, but it's good to see how different approaches can be generated from different people so that we can learn from one another. (Yes)

 

[kaliprasad]

Show that all real roots of the polynomial $f(x)=x^5-10x+38$ are negative.

Note:

I know this is a fairly easy challenge, but it's good to see how different approaches can be generated from different people so that we can learn from one another. (Yes)

Spoiler

$f(x)= x^5 + 10(3.8-x) $
it is >0 for $0\le x\lt3.8$

further
$f(x) = x(x^4-10) + 38$

for $x\gt 2$ above is > 0 as $2^4 = 16 \gt 10$

so above is >0 for $2\le x$

we have shown that it is positive for x > 0 so no root 0 or positive hence all real roots are -ve.

 

[anemone]

Spoiler

$f(x)= x^5 + 10(3.8-x) $
it is >0 for $0\le x\lt3.8$

further
$f(x) = x(x^4-10) + 38$

for $x\gt 2$ above is > 0 as $2^4 = 16 \gt 10$

so above is >0 for $2\le x$

we have shown that it is positive for x > 0 so no root 0 or positive hence all real roots are -ve.

Hi kaliprasad,

Thanks for participating.:) I am curious, is there any chance when you rewritten the function of $f$ as $f(x) = x(x^4-10) + 38$, you mean to imply the domain when $x\ge 10^{\tiny\dfrac{1}{4}}$?

 

[kaliprasad]

Hi kaliprasad,

Thanks for participating.:) I am curious, is there any chance when you rewritten the function of $f$ as $f(x) = x(x^4-10) + 38$, you mean to imply the domain when $x\ge 10^{\tiny\dfrac{1}{4}}$?

Spoiler

You are right but I have shown that it is true for x < 3.8 from the 1st equation and I have chosen a suitable value < 3.8 ( that is 2) to show that it is true for x > 2. $x\ge 10^{\tiny\dfrac{1}{4}}$ condition is a superset of it. but if it is true for x > 2 it does meet the criteria

 

[anemone]

Spoiler

You are right but I have shown that it is true for x < 3.8 from the 1st equation and I have chosen a suitable value < 3.8 ( that is 2) to show that it is true for x > 2. $x\ge 10^{\tiny\dfrac{1}{4}}$ condition is a superset of it. but if it is true for x > 2 it does meet the criteria

Oh My...I don't know what is wrong with me! How could I ask something so stupid! Sorry kali, I so wish to retract what I have asked just to appeared to be less silly...