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.