Let f ( x ) = x 5 − 5 x + p. Show that...

muroscamsey

muroscamsey

Answered

2022-08-12

Let f ( x ) = x 5 5 x + p. Show that f can have at most one root in [ 0 , 1 ],regardless of the value of p.

This seems to be an IVT problem, so I will go forth with that:

f ( a ) > u > f ( b )
f ( 0 ) > 0 > f ( 1 )
p > 0 > p 4
This is only true for p : ( 0 , 4 ) ie 0 < p < 4
Now I have proved that there is atleast one root between [ 0 , 1 ], now I need to prove that there isn't a second.

Would Rolles theorem be what I need here, or is there a simple deduction I can make from above to finish the problem?

Answer & Explanation

Hamza Conrad

Hamza Conrad

Expert

2022-08-13Added 20 answers

Yes, Rolle's theorem is perhaps the best tool for this. If f had two zeros, then the derivative
f ( x ) = 5 x 4 5 = 5 ( x 4 1 )
would vanish at a point in the interior of [ 0 , 1 ] - however, this isn't possible since x 4 1 is negative for all such x.
moiraudjpdn

moiraudjpdn

Expert

2022-08-14Added 2 answers

First you can see that f ( x ) = 5 ( x 4 1 ) is negative for the given interval. Therefore, the given function is a decreasing function. So, depending upon the value of the function at 0 and 1 it will have one or no roots. Hope this helps!

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