Domain restrictions from monotonicity

Questions asking to find parameter values or domain restrictions that ensure a function is monotonic (increasing/decreasing), typically using derivative sign analysis.

2 questions

CAIE P1 2016 November Q4

4 The function f is such that \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 2\) for \(x > n\), where \(n\) is an integer. It is given that f is an increasing function. Find the least possible value of \(n\).

CAIE P1 2019 November Q3

3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).