The gradient of a curve helps to identify if the functions are increasing curves or decreasing curves. Remember that the gradient of a line measures the rate of change of y with respect to the change in x. Hence it helps to identify the shape of a curve around a set of points.

We find the gradient of a tangent to the curve at a point – a positive gradient (/) represents an increasing curve (moving from left to right in x-axis), while a negative gradient (\) represents a decreasing curve at the point (moving from left to right in x-axis).

When the gradient of the tangent at a point on the curve is zero, we say that point to be stationary point and the tangent is parallel to the x-axis.

 

In summary,

• If f'(x) > 0, the curve is increasing (moving up)
• If f'(x) < 0, the curve is decreasing (moving down)
• If f'(x) = 0, the curve is stationary

A curve is monotonic increasing or decreasing if it is always increasing or decreasing (on either side of the stationary point); that is

• If f'(x) > 0, for all x, it is monotonic increasing
• If f'(x) < 0, for all x, it is monotonic deccreasing

 

Examples of increasing and decreasing curves

 

Example 1: Find all values of x for which the curve f(x) = x² + 3x + 2 is increasing

f(x) = x² + 3x + 2

f'(x) = 2x + 3

For the curve to be increasing, f'(x) > 0.

Therefore 2x + 3 > 0

or x >

 

Example 2: Find all the values of x for which the curve f(x) = x² + 5x + 4 is decreasing

f(x) = x² + 5x + 4

f'(x) = 2x + 5

For the curve to be increasing, f'(x) < 0.

Therefore 2x + 3 < 0

or x <

 

Example 3: Find the values of x for which the curve f(x) = x³ – x² – 4x + 4 is

• increasing
• decreasing
• stationary

f(x) = x³ – x² – 4x + 4

f'(x) = 3x² – 2x – 4

For the curve to be increasing, f'(x) > 0

f'(x) = 3x² – 2x – 4 > 0

Applying the quadratic equation x =

x = 

= 

= – 0.87

f'(x) > 0

(x – 1.54)(x + 0.87) > 0

x < -0.87 or x > 1.54

The curve is increasing for x < -0.87 and x > 1.54

For the curve to be decreasing, f'(x) < 0

f(x) = x³ – x² – 4x + 4

f'(x) = 3x² – 2x – 4 < 0

(x – 1.54)(x + 0.87) < 0

x > -0.87 or x < 1.54

The curve is decreasing for -0.87 < x < 1.54

For the stationary points, f'(x) = 0

f(x) = x³ – x² – 4x + 4

f'(x) = 3x² – 2x – 4 = 0

(x – 1.54)(x + 0.87) = 0

x = -0.87 and x = 1.54

Stationary points are –

When x = -0.87, y = (-0.87)³ – (-0.87)² – 4x(-0.87) + 4

= 6.06   (-0.87, 6.06)

When x = 1.54, y = (1.54)³ – (1.54)² – 4x(1.54) + 4

= 11.44   (1.54, 11.44)

Stationary points are (-0.87, 6.06) and (1.54, 11.44)

Here are some more examples of increasing and decreasing curves.