hey, again, can ne1 help wiv; find stationary points & if max/min of y=root(4x-1)+ 9/ root(4x-1)?

Answer 1

find stationary points & if max/min of

a) y=root(4x-1)+ 9/ root(4x-1) ;

with reservations 4x-1 > 0<=> x > 1/4

We see that we have to do with a composite function,
so in order to find the derivative we have to use the Chain Rule dy/dx=(dy/du)*(du/dx)
.....=========================

Let u(x)= 4x-1 and we are able to write the Equation a) as:

y(u) = u^(½)- 9*u^(-½) & u= 4x-1 =>

y'(u) = ½*u^(-½)+9*(-½)/u^(-3/2) & u'(x)= 4 =>

y´(x)=[ 1/(2*(4x-1)^(½)) - (9/2)/(4x-1)^(3/2)]*4 =
2/(4x-1)^(½) -18/[(4x-1)*(4x-1)^(½)] =
(2*(4x-1)-18)/(4x-1)^(½)
...........-------------------------------

Stationary points for y´(x) = 0
y´(x)=0 =>
2*(4x-1)-18 =0 <=> x= 20/8= 2½
...............................==========
Sign analysis for y'(x)

x=2½ => y'(2½)= 0
x=3 => y'(3) = 4
x=2 => y'(2) = -4

.......- - - - - 0 + + +
----nd---------l--------
---1/4-------2½------------>x

from the sign analysis we se the function is

Declining for 1/4 Growing for 2½< x
A rough draft can be made now:

**\***********/
***'\********/
******\ _ /
1/4---2,5-------->x
nd.......l..........................nd=not defined
conclusion :
Minimum for x=2,5 with fmin(2,5)= 6
===========================

Answer 2

y'= 1/2 (4x-1)^{-1/2} 4 - 9/2(4x-1)^{-3/2}

= [4(4x-1) -9]/(2(4x-1)^{3/2})
and it is =- if
4(4x-1) -9=0
so 16x-4-9=0
16x=13
x=13/16

if x=1, then y'>0
and if x=0, then y'<0
so the critical point is a (local) minimum

Answer 3

1) stationary points solve (for x) y' = 0

2) for each stat point detemine if this point is a max / min by looking at the sign of y' at the left side of stat point and at rightside of point + ^ - at ^ you have a max - ^ + you have a min,

weelll just look in your book.