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

Question

It is given that $\mathrm{f}\left( x \right)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$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 $\displaystyle\displaystyle \displaystyle\int_{\beta }^{\alpha }{\mathrm{f}\left( x \right)\mathrm{d}x},$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 $\mathrm{f}\left( x \right)=\mathrm{f}\left( -x \right).$What can be said about the six roots of the equation $\mathrm{f}\left( x \right)=0?$[4]

Accepted Answer

(i) $\alpha=1.885$ (3 d.p.), $\beta=\sqrt3$ (ii) $-0.597$ (3 d.p.) (iii) $\frac{54\sqrt3}{35}\text{ units}^2$ (iv) Two real roots $\pm\alpha$; the other four occur as $a\pm bi$ and $-a\pm bi$

N2014 P1 Q7

N2014 P1 Q7

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