We earlier saw that Stationary and turning points helped us to sketch a curve. It involved the following steps:
- Find x and y intercepts (where possible)
- Find where the curve increases or decreases by finding the derivative of the function
- Find the stationary points
- Sketch the graph through these points
In order to sketch the graph more accurately, we find the second derivative of the function. A second derivative is obtained by differentiating the function twice i.e first differentiating f(x) to get f'(x), and then differentiating f'(x) to get f”(x). Other notations for representing a second derivative is y”, or
Let’s take the function y = x3 – 2x2 + 4x + 3
= y’ = 3x2 – 4x + 4
= y” =
= 3.2x – 4 = 6x – 4
The sign of the second derivative gives us the information about the shape of a curve.
We observed in stationary points that f(x) is increasing when f'(x) > 0. f'(x) and f”(x) have a similar relationship i.e. f'(x) increases when f”(x) > 0. Conversely, f'(x) decreases when f”(x) < 0.
If f”(x) = 0, then f'(x) is stationary at that point.
If f”(x) > 0, then f'(x) is increasing, meaning the gradient of the tangent is increasing, and the curve is becoming steeper.
You will notice that the curve in both the diagrams are above the tangent, and the curve is shaping upwards, hence we say the curve is concave upwards.
On the other hand, if f”(x) < 0, then f'(x) is decreasing. Hence the value of the gradient of the tangent is decreasing, and the curve becomes less steep.
Here the curve in both the diagrams are below the tangent, and the curve is shaping downwards, hence we say the curve is concave downwards.
When f”(x) = 0, then f'(x) is stationary, i.e it neither increases nor decreases. At this point, the curve changes its concavity i.e if it has been going up, then the curve will begin to concave downwards after the stationary point. Similarly, if the curve was going down, then the curve will concave upwards after the stationary point. This point is also called the point of inflection.
Thus the shape of the curve can be found using the concavity of the curve.
When the curve is monotonic, the tangent is horizontal and the point of inflection is called the horizontal point of inflection. A curve has a point of inflection as long as the concavity changes at that point.
- If f”(x) > 0, the curve is concave upwards
- If f”(x) < 0, the curve is concave downwards
- If f”(x) = 0, and the concavity changes, there is a point of inflection (always check whether concavity changes around the point of inflection)
Now let’s look at some examples of concavity of a curve.