Increasing and decreasing Problem set B One question at a time

The graph shows y = f(x), a parabola with its lowest point at (−2, −3). Write the values of x for which f is increasing.

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

The graph shows y = f(x), a cubic with turning points at x = 0 and x = 3. Write down the values of x for which f is:

1. (a) increasing;

   • Turning points at x = 0 and x = 3 split the axis into three intervals
   • Where the graph rises, f is increasing
   • Increasing: \(0<x<3\)

2. (b) decreasing.

   • Where the graph falls, f is decreasing
   • Decreasing: \(x<0\) or \(x>3\)

The graph shows y = f(x) for 0 ≤ x ≤ 360, where x is in degrees. It has maxima at (0, 2) and (360, 2) and a minimum at (180, −4).

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

   • Minimum at x = 180; the graph is at a maximum at both ends
   • The graph rises from 180 to 360
   • Increasing: \(180<x<360\)

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 = 0, x = 180 and x = 360

The line shown is the tangent to the curve at P(1, 2). It passes through (0, 4) and (2, 0).

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

   • Gradient = rise ÷ run = (0 − 4) ÷ (2 − 0)
   • Gradient at P = −2

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 = 3
   • The tangent there is horizontal, so the gradient is 0 at x = 3

The graph shows the depth D centimetres of water in a tank t hours after 8 am, for 0 ≤ t ≤ 8. The turning points are (2, 8) and (6, 40).

1. (a) For which values of t is the depth of water decreasing?

   • Turning points at t = 2 and t = 6
   • Falling depth means D is decreasing
   • Decreasing: \(0\le t<2\) or \(6<t\le 8\)

2. (b) State the gradient of the graph at t = 6 and say what it tells you about the water.

   • The graph has a turning point at t = 6
   • Gradient 0: the depth is at its highest and is not changing at that moment

A function f is defined for all x. It is decreasing for x < 0, increasing for 0 < x < 4 and decreasing for x > 4. 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 = 0 and x = 4
• At x = 0 the graph changes from falling to rising: local minimum
• At x = 4 it changes from rising to falling: local maximum
• Falls, rises, falls: a cubic shape

The graph shows y = f(x). It is a quartic with two minima and one maximum.

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

   • Turning points at −3, 0, 3
   • The graph rises between the first minimum and the maximum, and to the right of the last minimum
   • Increasing: \(-3<x<0\) or \(x>3\)

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

   • The gradient is zero at every turning point
   • x = −3, x = 0, x = 3