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

$y=\sin(x^4)$ $\implies y'=4x^3\cos(x^4)$ $\implies y'=12x^2\cos(x^4)-16x^6\sin(x^4)$ 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 $-\dfrac\pi2+2n\pi\cos x$ for $\pi\sin x$ for [latex]\dfrac{5\pi}4