Show that the function F(x)=(x-a)^2 (x-b)^2 +x has the value (a+b)/2 at some point x.

Dean Summers

Dean Summers

Answered question

2022-07-27

Show that the function F ( x ) = ( x a ) 2 ( x b ) 2 + x has the value (a+b)/2 at some point x.

Answer & Explanation

sweetwisdomgw

sweetwisdomgw

Beginner2022-07-28Added 20 answers

It is obvious to say that F(x) is continuous as it is just apolynomial of order 4.
And Now define G(x)=F(x)-(a+b)/2
G(x) is a polynomial of order 4,Henceit is also continuous .
Now find G(a) and G(b),
G ( a ) = F ( a ) ( a + b ) / 2 = ( a b ) / 2
and G(b)=F(b)-(a+b)/2=(b-a)/2
G ( a ) = G ( b )
i.e G(a) and G(b) have opposite signs and it is continuoustoo.
Hence the curve G(x) must cut the x-axis at some pointα. i.e G(a)=0
G ( a ) = F ( a ) ( a + b ) / 2 = 0
F ( a ) = ( a + b ) / 2

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