Is the graph of y = sin(x^4) increasing or decreasing when x = 10? Is it concave up or concave down?
(1) Answers
[latex]y=\sin(x^4)[/latex]
[latex]\implies y'=4x^3\cos(x^4)[/latex]
[latex]\implies y'=12x^2\cos(x^4)-16x^6\sin(x^4)[/latex]
At [latex]x=10[/latex], you have
[latex]y'(10)=4000\cos(10^4)[/latex]
The trick to finding out the sign of this is to figure out between which multiples of [latex]\dfrac\pi2[/latex] the value of [latex]10^4[/latex] lies.
We know that [latex]\cos x>0[/latex] whenever [latex]-\dfrac\pi2+2n\pi
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