It is given that $\mathrm{f}\left( x \right)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve wi

Question

It is given that $\mathrm{f}\left( x ight)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve with equation $y=\mathrm{f}\left( x ight)$and the line with equation$y=-7,$for $x\ge 0.$ The curve crosses the positive $x$-axis at $x=\alpha ,$and the curve and the line meet where $x=0$ and $x=\beta .$ Find the value of $\alpha ,$giving your answer correct to $3$ decimal places, and find the exact value of $\beta .$ [2] Evaluate $\displaystyle\int
olimits_{\beta }^{\alpha }{\mathrm{f}\left( x ight)\mathrm{d}x},$giving your answer correct to $3$ decimal places. [2] Find, in terms of $\sqrt{3},$the area of the finite region bounded by the curve and the line, for $x\ge 0.$ [3] Show that $\mathrm{f}\left( x ight)=\mathrm{f}\left( -x ight).$What can be said about the six roots of the equation $\mathrm{f}\left( x ight)=0?$ [4]

Accepted Answer

(i) α = 1.885, β = √3; (ii) −0.597; (iii) (54/35)√3 units²; (iv) roots: α, −α, a±bi, −a±bi

N2014 P1 Q7

N2014 P1 Q7

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