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2026 NYGH P1 Q8

2026 NYGH P1 Q8

IP 4
9 marks

The diagram shows part of the curve \(y=\dfrac12x^3-2x+3\) for \(-3.5\le x\le3.5\).

  1. Use your graph to find the range of values of \(x\) for which \(\dfrac12x^3-2x+3\le3\).[2]
  2. A line \(y=\dfrac32x+c\), where \(c>0\), is a tangent to the curve \(y=\dfrac12x^3-2x+3\) at point \(P\).
    By drawing a suitable straight line on the same axes, use your graph to find
    1. the coordinates of point \(P\),[2]
    2. the value of \(c\).[1]
  3. By adding another suitable straight line on the same axes, solve the equation \(x^3-3x-24=0\).[4]

Solution:

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Answer:(a) \(-3.5\le x\le-2\) or \(0\le x\le2\) (b)(i) \(P\approx(-1.53,4.27)\) (ii) \(c\approx6.56\) (c) \(x\approx3.23\)

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