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 2022-07-27 Answered
Show that the function F ( x ) = ( x a ) 2 ( x b ) 2 + x has the value (a+b)/2 at some point x.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

sweetwisdomgw
Answered 2022-07-28 Author has 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

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-09-07

Determine whether each of these functions is a bijection from R to R.

a) f(x)=3x+4

b) f(x)=3x2+7

c) f(x)=x+1x+2
d)f(x)=x5+1

asked 2021-09-10
A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at 10, the average attendance had been 27,000. When ticket prices were lowered to10,the average attend ance had been 27,000.When ticket prices were lowered to 8, the average attendance rose to 33,000. How should ticket prices be set to maximize revenue?
asked 2021-06-11

Find the linear approximation of the function f(x)=1x at a=0 and use it to approximate the numbers 0.9 and 0.99.

asked 2021-08-17
Match the function with its name: f(x) = c
a) squaring function
b) square root function
c) cubic function
d) linear function
e) constant function
f) absolute value function
g) greatest integer function
h) reciprocal function
i) identity function
asked 2021-12-06
Two waves in one string are described by the wave functions:
y1=(3.0cm)cos(4.0×1.6t)
y2=(4.0cm)sin(5.0×2.0t)
where y and x are in centimeters and t is inseconds. Find the superposition of the waves y1+y2 at the points (a) x=1.00, t=1.00; (b)x=1.00, t=.500; and (c) x= 0.500, t=0.
asked 2022-01-19
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form. 2, 3+2i, sqrt 5
asked 2022-09-03
Find a function that models the surface area S of a cube in terms of its volume.

New questions