Question

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

Answer 1

$y=\sin(x^4)$
$\implies y'=4x^3\cos(x^4)$
$\implies y'=12x^2\cos(x^4)-16x^6\sin(x^4)$

At $x=10$, you have

$y'(10)=4000\cos(10^4)$

The trick to finding out the sign of this is to figure out between which multiples of $\dfrac\pi2$ the value of $10^4$ lies.

We know that $\cos x>0$ whenever $-\dfrac\pi2+2n\pi$, and that $\cos x$ whenever $\dfrac\pi2+2n\pi$, where $n\in\mathbb Z$.

We have

$10^4=\dfrac{k\pi}2\implies k=\dfrac{2\times10^4}\pi\approx6366.2$

which is to say that $\dfrac{6366\pi}2$, an interval that is equivalent modulo $2\pi$ to the interval $\left(\pi,\dfrac{3\pi}2\right)$.

So what we know is that $10^4$ corresponds to the measure of an angle that lies in the third quadrant, where both cosine and sine are negative.

This means $y'(10)$, so $y$ is decreasing when $x=10$.

Now, the second derivative has the value

$y'=12\times10^2\cos(10^4)-16\times10^6\sin(10^4)$

Both $\cos(10^4)$ and $\sin(10^4)$ are negative, so we're essentially computing the sum of a negative number and a positive number. Given that $\sin x>\cos x$ for $\pi$, and $\cos x>\sin x$ for $\dfrac{5\pi}4$, we can use a similar argument to establish in which half of the third quadrant the angle $10^4$ lies. You'll find that the sine term is much larger, so that the second derivative is positive, which means $y$ is concave up when $x=10$.