x=4cost , y= 2sint , 0

x=4cost
y=2sint
First, take the derivative of x and y with respect to t.
dx/dt=-4sint
dy/dt=2cost
Then, determine the first derivative dy/dx . Take note that in parametric equation, the formula of dy/dx is:
dy/dx= (dy/dt)/(dx/dt)
Applying this formula, the first derivative is:
dy/dx= (2cost)/(-4sint)
dy/dx=-1/2cott
Then, determine the second derivative of the parametric equation. To do so, apply the formula:
(d^2y)/(dx^2)= (d/dt (dy/dx))/(dx/dt)
So the second derivative is:
(d^2y)/(dx^2) = (d/dt(-1/2cott))/(-4sint)
(d^2y)/(dx^2) = (1/2csc^2t)/(-4sint)
(d^2y)/(dx^2)=-1/8csc^3t
Take note that the concavity of the curve changes when the second derivative is zero or does not exist.
(d^2y)/(dx^2)= 0   or   (d^2y)/(dx^2)= DNE
Setting the second derivative equal to zero result to no solution.
-1/8csc^3t = 0
t={O/}
Since there are no angles in which cosecant will be zero.
However, on the interval 0-1/8csc^3t= DNE
t=pi
So the concavity of the parametric curve changes at t=pi .
Now that the inflection is known, apply the second derivative test.
Take note that when the value of the second derivative on an interval is positive, the curve on that interval is concave up.
(d^2y)/(dx^2)gt0     :. concave up
And when the value of the second derivative on an interval is negative, the curve on that interval is concave down.
(d^2y)/(dx^2)lt0     :. concave down
So divide the given interval 0lttlt2pi into two regions. The regions are 0lttltpi and pilttlt2pi . Then, assign a test value for each region. And, plug-in the test values to the second derivative.
For the first region 0lttltpi , let the test value be t=pi/2 .
(d^2y)/(dx^2)=-1/8csc^3(pi/2) = -1/8(1)^3=-1/8
So the parametric curve is concave down on the interval 0lttltpi .
For the second region pilttlt2pi , let the test value be t=(3pi)/2 .
(d^2y)/(dx^2)=-1/8csc^3((3pi)/2) = -1/8(-1)^3=-1/8*(-1)=1/8
So it is concave up on the interval pilttlt2pi .
 
Therefore, the graph of the given parametric equation is concave down on the interval 0lttltpi and it is concave up on the interval pilttlt2pi .