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

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

kinggeorge · Answer

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

anonymous · Answer

im will guess mvt for this one
am i right?

kinggeorge · Answer

That's what I was thinking as well.

anonymous · Answer

oh maybe not. hmm we have 
$$F(a)=a$$
$$F(b)=b$$

anonymous · Answer

no its under intermediate value theoremm

anonymous · Answer

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

anonymous · Answer

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 $b$

kinggeorge · Answer

Looks right. And if $b<a$ $$b<{a+b\over2}<a$$

anonymous · Answer

perfectt thanks alot guyss