geo calculus question (1 Viewer)

darshil · Mar 14, 2009

Find the stationary points of each of the following curves and use the second derivative to determine their nature
(b) y=1-x^3

I diffrentiated it and made it equal to zero for the stationary points but im stuck at it
i get all the other types, its just this one

i bet its way simple and i cant click on it

thanks heaps !

cutemouse · Mar 14, 2009

Well firstly, look at the equation it's a cubic with it's inflexion (or "vertex" if you like) at (0,1) and it's concavity is opposite to the basic cubic equation (ie. y=x³), So that should give you a hint of what to expect, etc.

y=1-x³
y'=-3x²
y''=-6x

For a stat point y'=0
-3x²=0
x=0

When x=0 y''=0 => No new information. Use the 1st deriv sign test.

When x=0^- y'<0
When x=0⁺ y'<0 [Would ideally use a table for this]

Code:

\__
   \

Therefore horizontal point of inflexion at (0,1)

kurt.physics · Mar 14, 2009

darshil said:
Find the stationary points of each of the following curves and use the second derivative to determine their nature
(b) y=1-x^3

I diffrentiated it and made it equal to zero for the stationary points but im stuck at it
i get all the other types, its just this one

i bet its way simple and i cant click on it

thanks heaps !

You just need to remember the types of stationary (turning) points

$$-maximum: when$\,\,\frac{dy}{dx} = 0\,\,$and$\,\,\frac{d^2y}{dx^2}<0$

$$-minimum: when$\,\,\frac{dy}{dx} = 0\,\,$and$\,\,\frac{d^2y}{dx^2}>0$

$$-inflexion: when concavity changes ie$\,\,\frac{d^2y}{dx^2}=0$

$Now a horizontal inflexion point is special in the sense that$\,\,\frac{dy}{dx}=0\,\,$and$\,\,\frac{d^2y}{dx^2}=0.\,\,\,$But it is only a possible point of inflextion, so you have to check that concavity changes with the second derivative.$

sinophile · Mar 14, 2009

darshil said:
Find the stationary points of each of the following curves and use the second derivative to determine their nature
(b) y=1-x^3

I diffrentiated it and made it equal to zero for the stationary points but im stuck at it
i get all the other types, its just this one

i bet its way simple and i cant click on it

thanks heaps !

For stat. pts, let first deriative equal to zero and solve for x. nature is determined by second derivative: sign determines concavity. if it tests as zeo in the 2nd derivative, tetsing of immediate region needs to happen because you canot be sure of its nature atm.

y=1-x^3

y'=-3x^2

y''=-6x

if y'=0

-3x^2=0
x=0

at x=0
f(x)=1
f''(x)=0

test region aroound x=0:
x -1 0 1
y'' + 0 -

therefore 0,0 is a pt of inflexion

edit: i see i did a silly mistake. fuck

geo calculus question (1 Viewer)

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