Question

How do you show that the Mean Value theorem applies to `f(x) = xsqrt(6-x)` on the interval `[0,6]`, and find a value of `c` guaranteed by this theorem?

Answer 1

`c=1.946654` or `5.490467`

Explanation:

Mean Valuetheorem states that a continuous function on a closed, bounded interval has at least one point where it is equal to its average value on the interval.

Mathematically speaking if a function `f(x)` is a continuous function on the closed, bounded interval `[a,b]`, then there is at least one number `c` in `(a,b)` for which

`f(c)=1/(b-a)int_a^bf(x)dx`

Here `f(x)=xsqrt(6-x)` is continuous over `[0,6]`

to workout `c`, let us work out `int_0^6xsqrt(6-x)dx`

Assume `u=x` then `du=dx` and `v=(6-x)^(3/2)` then `dv=-3/2sqrt(6-x)dx`

Now using integration by parts as `intudv=uv-intvdu`, we have

`-3/2intxsqrt(6-x)dx=x(6-x)^(3/2)-int(6-x)^(3/2)dx`

or `-3/2intxsqrt(6-x)dx=x(6-x)^(3/2)-(-2/5(6-x)^(5/2))`

or `intxsqrt(6-x)dx=-2/3[x(6-x)^(3/2)+2/5(6-x)^(5/2)]`

= `-2/3(6-x)^(3/2)[x+2/5(6-x)]=-2/15(6-x)^(3/2)[5x+12-2x]`

= `-2/5(6-x)^(3/2)(x+4)`

and `int_0^6xsqrt(6-x)dx`

= `-2/5[(6-x)^(3/2)(x+4)]_0^6`

= `2/5[6^(3/2)xx4]=8/5*6^(3/2)=48/5sqrt6`

and `f(c)=48/5sqrt6/6=(8sqrt6)/5`

or `csqrt(6-c)=(8sqrt6)/5`

or `25c^2(6-c)=384`

or `25c^3-150c^2+384=0`

and using Goal seek in MSExcel, we get

`c=1.94665399787863` or `5.49046652705594`

Answer 2

The value of `c=4`

Explanation:

The mean value theorem states that if `f(x)` is a continuous on the interval `I=[a,b]` and differentiable on the interval `(a,b)`, then
there exists `c in (a,b)` such that

`f'(c)=(f(b)-f(a))/(b-a)`.

Here,

`f(x)=xsqrt(6-x)`

`f(x)` is the product of continuous and differentiable functions

The domain of `f(x)` is `D_(f(x))=(-oo,6]`

`I=[0,6]`

`f(0)=0*sqrt6=0`

`f(6)=6*sqrt(6-6)=6*0=0`

Therefore,

`f'(x)=sqrt(6-x)-x/(2sqrt(6-x))=(2(6-x)-x)/(2sqrt(6-x))`

`=(12-3x)/(2sqrt(6-x))`

Then,

`EE c` such that

`(12-3c)/(2sqrt(6-c))=(f(6)-f(0))/(6-0)=0`

`12-3c=0`

`c=4`

This is also, Rolle's theorem.

graph{xsqrt(6-x) [-5.12, 14.88, -2, 8]}