c3 question-stationary point

curve with equation y=x^(5/2)ln(x/4) crosses x-axis at (4,0)

the curve has a stationary point at R

find the x coordinate of R in exact form?

Reply 1

Dekota

jenh1

curve with equation y=x^(5/2)ln(x/4) crosses x-axis at (4,0)

the curve has a stationary point at R

find the x coordinate of R in exact form?

y = x^5/2ln(x/4)
y = (x⁵)^1/2ln[x/4]
y = √x⁵ln[x/4]
dy/dx = d[√x⁵]/dx.ln[x/4] - d[ln[x/4]]/dx.√x⁵
....... = 5/2x^3/2ln[x/4] - 1/x√x⁵
....... = 5/2√x³ln[x/4] - 1/x√x⁵
....... = √x³(5/2ln[x/4] - 1)

sub (x = 4) into dy/dx ...
....... = √4³(5/2ln[4/4] - 1)
....... = √64(5/2ln1 - 1)
....... = -8

Reply 2

Dekota

jenh1

curve with equation y=x^(5/2)ln(x/4) crosses x-axis at (4,0)

the curve has a stationary point at R

find the x coordinate of R in exact form?

y = x^5/2ln(x/4)
y = (x⁵)^1/2ln[x/4]
y = √x⁵ln[x/4]
dy/dx = d[√x⁵]/dx.ln[x/4] - d[ln[x/4]]/dx.√x⁵
....... = 5/2x^3/2ln[x/4] - 1/x√x⁵
....... = 5/2√x³ln[x/4] - 1/x√x⁵
....... = √x³(5/2ln[x/4] - 1)

sub (dy/dx = 0) ...
....... => √x³(5/2ln[x/4] - 1) = 0

x = 0? not sure

Reply 3

steve2005

jenh1

curve with equation y=x^(5/2)ln(x/4) crosses x-axis at (4,0)

the curve has a stationary point at R

find the x coordinate of R in exact form?

Here is my solution.

Reply 4

random123

To find dy/dx:

Using product rule:

y = x^(5/2)ln(x/4)
y = uv

where u = x^(5/2)
and v = ln(x/4)

du/dx = (5/2)x^(3/2)

and find dv/dx using the chain rule:

dv/dx = dv/dw x dw/dx where w = x/4

dv/dw = 1/w
dw/dx = 1/4

dv/dx = 4/x mulitplied by 1/4
= 1/x

Back to product rule:

dy/dx = uv' + vu'
= x^(5/2).(1/x) + (ln(x/4)).(5/2)x^(3/2)
= x^(3/2) + ln(x/4).(5/2)x^(3/2)

Then you say dy/dx = 0 and go from there.

I'm pretty sure it's possible from there but I've forgotten how to do logs. :biggrin:

Reply 5

carol05

Joe_87

Then you say dy/dx = 0 and go from there.

I'm pretty sure it's possible from there but I've forgotten how to do logs. :biggrin:

The question asks for the exact value of the x co-ordinate of R. Steve has answered the question.

Reply 6

random123

carol05

The question asks for the exact value of the x co-ordinate of R. Steve has answered the question.

Yes I do know. Well done to steve. He happened to finish his post before I did, otherwise I wouldn't have bothered with my answer, which I am fully aware is incomplete.