A polynomial function P(x) with degree 5 increases in the interval (-infty, 1) and (3, infty) and decreases in the interval (1,3). Given that P′(2)=0 and P(0)=4, find P′(6).

memLosycecyjz 2022-09-15 Answered
Why is 2 double root of the derivative?
A polynomial function P(x) with degree 5 increases in the interval ( , 1 ) and ( 3 , ) and decreases in the interval (1,3). Given that P ( 2 ) = 0 and P ( 0 ) = 4, find P'(6).
In this problem, I have recognised that 2 is an inflection point and the derivative will be of the form: P ( x ) = 5 ( x 1 ) ( x 3 ) ( x 2 ) ( x α ).
But I am unable to understand why x = 2 is double root of the derivative (i.e. why is α = 2?). It's not making sense to me. I need help with that part.
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Answers (2)

Harold Beltran
Answered 2022-09-16 Author has 3 answers
Step 1
2 is an inflexion point so necessarily P ( 2 ) = 0.
Step 2
Now P ( 2 ) = P ( 2 ) = 0 so 2 is a double root of P′.

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Averi Fields
Answered 2022-09-17 Author has 3 answers
Explanation:
If 2 is an inflexion point then it is necessary that P ( 2 ) = 0, which means that you have a factor of ( x 2 ) 2 in P′(x).

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