I've an homework question where i need to prove that the following equation contains at-least three roots x 4 10 = x 4 − 100 x − 1 .I was able to find three roots after blackefining the equation as function: f ( x ) = x 4 − 100 x − 1 − x 4 10 where the segments:1. [3.3,3.4] because f ( 3.3 ) &gt; 0 and f ( 3.4 ) &lt; 02. [10.9,11] because f ( 10.9 ) &gt; 0 and f ( 11 ) &lt; 03. [−3,−2] because f ( − 3 ) &lt; 0 and f ( − 2 ) &gt; 0I know the Intermediate-Value theorem state that for each mentioned segment there has to be a 0 (a root).I would like to know if my answer is correct and if so how can i prove that the function is continuous at the mentioned segments.UPDATE: The roots are indeed in the above mentioned segments but in the open segment like:1. (3.3,3.4)2. (10.9,11)3. (−3,−2)

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I&#039;ve an homework question where i need to prove that the following equation contains at-least three roots                     x        4            10        =                              x          4                −        100                    x        −        1            .I was able to find three roots after blackefining the equation as function:   f  (  x  )  =                              x          4                −        100                    x        −        1              −                    x        4            10       where the segments:1. [3.3,3.4] because   f  (  3.3  )  &amp;gt;  0 and   f  (  3.4  )  &amp;lt;  02. [10.9,11] because   f  (  10.9  )  &amp;gt;  0 and   f  (  11  )  &amp;lt;  03. [−3,−2] because   f  (  −  3  )  &amp;lt;  0 and   f  (  −  2  )  &amp;gt;  0I know the Intermediate-Value theorem state that for each mentioned segment there has to be a 0 (a root).I would like to know if my answer is correct and if so how can i prove that the function is continuous at the mentioned segments.UPDATE: The roots are indeed in the above mentioned segments but in the open segment like:1. (3.3,3.4)2. (10.9,11)3. (−3,−2)

Joe Sheppard · Accepted Answer

Assuming   x  ≠  1, the equation you have to solve is      x    5    −  11      x    4    +  1000  =  0So, consider the function and derivatives  f  (  x  )  =      x    5    −  11      x    4    +  1000      f    ′    (  x  )  =  5      x    4    −  44      x    3        f    ″    (  x  )  =  20      x    3    −  132      x    2  The first derivative cancels at   x  =  0 and   x  =      44    5  .  f  (      44    5    )  =  −      38104056    3125    &amp;lt;  0and the second derivative test shows that this is a minimum; so you have two real roots.If you deflate the quintic polynomial, you are let with a cubic which has at least one real root.So, you have at least three real roots that you properly located.

Massatfy · Accepted Answer

Hint function is continuous at mentioned points if there exists a limit at that point and its value is equal to the value of the function at that point

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