Increasing and decreasing Problem set A Worked solutions

The graph shows y = f(x), a parabola with its highest point at (2, 4). Write the values of x for which f is increasing.

• Vertex at x = 2: gradient zero
• The graph rises up to the vertex and falls after it
• Increasing: \(x<2\)

The graph shows y = f(x), a cubic with turning points at x = −2 and x = 1. Write down the values of x for which f is:

1. (a) increasing;

   • Turning points at x = −2 and x = 1 split the axis into three intervals
   • Where the graph rises, f is increasing
   • Increasing: \(x<-2\) or \(x>1\)

2. (b) decreasing.

   • Where the graph falls, f is decreasing
   • Decreasing: \(-2<x<1\)

The graph shows y = f(x) for 0 ≤ x ≤ 360, where x is in degrees. It has a maximum at (90, 4) and a minimum at (270, −4).

1. (a) Write down the values of x for which f is increasing.

   • Maximum at x = 90, minimum at x = 270
   • The graph rises from 0 to 90 and from 270 to 360
   • Increasing: \(0\le x<90\) or \(270<x\le 360\)

2. (b) Write down the values of x for which the gradient of the graph is zero.

   • Gradient is zero at the turning points
   • x = 90 and x = 270

The line shown is the tangent to the curve at P(7, 9). It passes through (4, 0) and (6, 6).

1. (a) Find the gradient of the curve at P.

   • Gradient = rise ÷ run = (6 − 0) ÷ (6 − 4)
   • Gradient at P = 3

2. (b) Write down the value of x at which the gradient of the curve is zero.

   • The lowest point of the curve is at x = 1
   • The tangent there is horizontal, so the gradient is 0 at x = 1

The graph shows the height h metres of a cyclist above the valley floor after riding d kilometres along a route, for 0 ≤ d ≤ 6. The turning points are (2, 20) and (4, 16).

1. (a) For which values of d is the cyclist going uphill?

   • Turning points at d = 2 and d = 4
   • Uphill means h is increasing
   • Uphill: \(0\le d<2\) or \(4<d\le 6\)

2. (b) State the gradient of the graph at d = 4 and say what it tells you about the road.

   • The graph has a turning point at d = 4
   • Gradient 0: the cyclist is neither climbing nor descending at that moment; the road is level

A function f is defined for all x. It is increasing for x < −1, decreasing for −1 < x < 3 and increasing for x > 3. Describe the graph of y = f(x): where the gradient is zero, what kind of turning points they are, and the overall shape.

• Gradient changes sign only at turning points: x = −1 and x = 3
• At x = −1 the graph changes from rising to falling: local maximum
• At x = 3 it changes from falling to rising: local minimum
• Rises, falls, rises: a cubic shape

The graph shows y = f(x), a quartic with a W shape.

1. (a) Write down the values of x for which f is increasing.

   • Turning points at −2, 0, 2
   • The graph rises between the first minimum and the maximum, and to the right of the last minimum
   • Increasing: \(-2<x<0\) or \(x>2\)

2. (b) Write down the values of x at which the gradient is zero.

   • The gradient is zero at every turning point
   • x = −2, x = 0, x = 2