Is the graph of y = sin(x^4) increasing or decreasing when x = 10? is it concave up or concave down?

At , you have



The trick to finding out the sign of this is to figure out between which multiples of the value of lies.

We know that whenever , and that whenever , where .

We have



which is to say that , an interval that is equivalent modulo to the interval .

So what we know is that corresponds to the measure of an angle that lies in the third quadrant, where both cosine and sine are negative.

This means , so is decreasing when .

Now, the second derivative has the value



Both and are negative, so we're essentially computing the sum of a negative number and a positive number. Given that for , and for , we can use a similar argument to establish in which half of the third quadrant the angle lies. You'll find that the sine term is much larger, so that the second derivative is positive, which means is concave up when .