Complete the table of values for \(y=\dfrac15x^3-x^2+1\).[1]
\(x\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(y\)
\(-4.6\)
\(-0.2\)
\(1\)
\(0.2\)
\(-2.6\)
\(-2.2\)
\(1\)
On the grid provided, draw the graph of \(y=\dfrac15x^3-x^2+1\) for \(-2\leq x\leq5\).[3]
On the same grid, draw the line \(y=\dfrac12x-1\) for \(-2\leq x\leq5\).[1]
Write down the \(x\)-coordinates of the points where the line intersects the curve.[2]
Show that these \(x\)-values are the solutions of the equation \(2x^3-10x^2-5x+20=0\).[2]
The equation \(\dfrac15x^3-x^2+1=k\) has two solutions.
Use your graph to find the two possible values of \(k\).[2]
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Answer:(a) \(-1.4\) (b) See graph (c)(i) See line (ii) \(x\approx-1.45,1.35\) within the plotted interval (iii) \(2x^3-10x^2-5x+20=0\) (d) \(k=1\) or approximately \(-2.7\).