Prove or show increasing/decreasing function

A question is this type if and only if it asks to prove, show, or verify that a given function is always increasing, always decreasing, or never negative/positive in gradient, typically by showing the derivative has a fixed sign for all x.

4 questions · Moderate -0.8

1.07o Increasing/decreasing: functions using sign of dy/dx

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CAIE P1 2013 June Q1

3 marks Moderate -0.8

1 It is given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ^ { 3 } + x\), for \(x \in \mathbb { R }\). Show that f is an increasing function.

CAIE P1 2011 November Q2

3 marks Moderate -0.8

2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.

CAIE P1 2012 November Q2

3 marks Moderate -0.8

2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.

OCR C1 Q2

4 marks Moderate -0.8

2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).