Find dy/dx and d2y/dx2. x = cos(2t), y = cos(t), 0 < t < π dy dx = correct: your answer is correct. d2y dx2 = correct: your answer is correct. for which values of t is the curve concave upward? (enter your answer using interval notation.) incorrect: your answer is incorrect.

The value of 't' at which the curve is concave upward is and this can be determined by using the properties of the trigonometric function.

Given :

First, differentiate 'x' with respect to 't'.

Now, differentiate 'y' with respect to 't'.

Now, the value of dy/dx is given by:

Now, the expression of is given by:

    --- (1)

where f is the function of 't'.

Substitute the value of df/dt in the expression (1).

For the curve to be concave upward the expression of the second derivative should be greater than zero.

Now, the value of 't' at which (cos t < 0) is given by:

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By the chain rule, the first derivative is

We have

so that

The second derivative is

where is a function of ; denote it by . Then by the chain rule,

Since

we end up with

The curve is concave upward whenever the second derivative is positive. We have this for

We have for where is any integer; for the given interval, this happens at . For , we have , while for , we have ; it's the latter that we care about, since over this interval we have .

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