Mathematics
Santelo
25

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

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(1) Answers
CollinsLeeHinch

[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\cos x[/latex] for [latex]\pi\sin x[/latex] for [latex]\dfrac{5\pi}4

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