Question

Show that the function F(x)=(x-a)^2 * (x-b)^2 + x takes on some value (a+b)/2 for some value x

please helpp

Answer 1

So we want to show that \[(x-a)^2 \cdot (x-b)^2 + x={a+b \over 2}\]for some value of x.

Answer 2

im will guess mvt for this one
am i right?

Answer 3

That's what I was thinking as well.

Answer 4

oh maybe not. hmm we have
\[F(a)=a\]
\[F(b)=b\]

Answer 5

no its under intermediate value theoremm

Answer 6

so maybe it is intermediate value theorem, since if \(a<b\) we have
\[a< \frac{a+b}{2}<b\]

Answer 7

ok so done
F is continuous, must take on all values between \(F(a)\) and \(F(b)\) so must take on all values between \(a\) and \(b\)

Answer 8

Looks right. And if \(b<a\) \[b<{a+b\over2}<a\]

Answer 9

perfectt thanks alot guyss