[mathjax]
Let’s have a look at another question from Methodist Girls School
(MGS Prelim 2016 AMath Paper 2 Q 1b)
This question is in line with the 4047 Syllabus where students have to know
“Conditions for [latex]ax^2+bx+c[/latex] to be always positive (or always negative)”
Above extracted from SEAB.
Question: Determine the conditions for [latex]p[/latex] and [latex]q[/latex] such that the curve [latex]y=px^2-2x+3q[/latex] lies entirely above the[latex]x[/latex]axis, where [latex]p[/latex] and [latex]q[/latex] are constants. [3 marks]
Solution: The keyword is “entirely above”, implying that the graph is above the axis. We expect a Happy Face that is not intersecting the x-axis.
[latex]\Rightarrow[/latex] Happy Face is ABOVE axis
[latex]\Rightarrow[/latex] No Real Solutions
[latex]\Rightarrow[/latex] [latex]b^2-4ac < 0[/latex]
Observe that [latex]a=p[/latex], [latex]b=-2[/latex] and [latex]c=3q[/latex],
[latex]b^2-4ac < 0 [/latex]
[latex](-2)^2 – 4(p)(3q) < 0 [/latex]
[latex]4-12pq<0[/latex]
[latex]1-3pq<0[/latex]
[latex]\frac{1}{3} < pq [/latex]
Since the graph is a happy face, we know that the coefficient of [latex]x^2[/latex] is positive and [latex]p>0[/latex].
Therefore, [latex] p>0[/latex] and [latex]pq > \frac{1}{3} [/latex]