Curve above or below axis

Show that a curve lies entirely above or below the x-axis by analyzing the discriminant or minimum/maximum value.

3 questions · Standard +0.2

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CAIE P1 2005 June Q10
10 marks Standard +0.3
10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).
  1. Show that the whole of the curve lies above the \(x\)-axis.
  2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
  3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
  4. Find the value of \(k\) for which the line is a tangent to the curve.
CAIE P1 2018 June Q2
5 marks Moderate -0.5
2 The equation of a curve is \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis.
  2. Find the value of \(k\) for which the line \(y + 2 x = 7\) is a tangent to the curve.
OCR FP2 2009 June Q2
4 marks Standard +0.8
2 Given that \(y = \frac { x ^ { 2 } + x + 1 } { ( x - 1 ) ^ { 2 } }\), prove that \(y \geqslant \frac { 1 } { 4 }\) for all \(x \neq 1\).