Second Derivative and Concavity of Curves

 

A function can be differentiated several times.

Differentiating f(x) gives f'(x).

Differentiating f'(x) gives f''(x)

Differentiating f''(x) gives f'''(x)

Some other notations (similar meaning) are:   \cfrac{{d}^{2}y}{d{x}^{2}} \quad , \quad y''   and so on...

 

Examples

 

1. Find the second derivative of y={(2x+5)}^{7}.

Show Answer

\cfrac { dy }{ dx } =7{ (2x+5) }^{ 6 }\cdot 2\\ \\ \cfrac { dy }{ dx } =14{ (2x+5) }^{ 6 }\\ \\ \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =14\cdot 6{ (2x+5) }^{ 5 }\cdot 2\\ \\ \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =168{ (2x+5) }^{ 5 }

 

 

2. Find f''(-1) if f(x)={x}^{4}-1.

Show Answer

f'(x)=4{x}^{3} \\ f''(x)=12{x}^{2} \\ \therefore \quad f''(-1)=12{(-1)}^{2} = 12

 

 

Concavity

 

The sign of the second derivative shows information about the curve's shape:

if f"(x)>0 then f'(x) is increasing

if f"(x)<0 then f'(x) is decreasing

if f"(x)=0 then f'(x) is stationary

 

1. Increasing Gradient

If f"(x)>0 then f'(x) is increasing. This means that the gradient of the tangent kept on increasing and the curve becomes steeper, and the curve is concave upwards.

 

2. Decreasing Gradient

If f"(x)<0 then f'(x) is decreasing. This means that the gradient of the tangent kept on decreasing and the curve becomes less steep, and the curve is concave downwards.

 

3. Stationary Points

If f"(x)=0 then f'(x) is stationary. This means there is a change of the curve's concavity: from being concave upwards to downwards OR from being concave downwards to upwards. The curve goes through a changing concavity and has a point of inflexion.

 

Use the Geogebra app below - move point D to understand the concavity at that particular point. Click on the f'(x) and f''(x) checkbox to show the derivative and the second derivative. Have fun!

 

 

- Jesse Parete, 2012

 

Therefore, we can conclude that:

 

Examples

 

1. Find the point of inflexion on the curve y={x}^{3}-6{x}^{2}+5x+12

Show Answer

y'=3{x}^{2}-12x+5 \\ y''=6x-12

For inflexions,

y''=0 \\ 6x-12=0 \\ 6x=12 \\ x=2

When x=2,

y={2}^{3}-6{(2)}^{2}+5(2)+12=6

Hence, (2, 6) is a possible point of inflexion. We check the concavity changes:

 

Since concavity changes, (2, 6) is a point of inflexion.

 

2. Find all the values of x or which the curve f(x)=2{x}^{3}-7{x}^{2}-5x+6 is: (i) concave downwards, and (ii) concave upwards.

Show Answer

f'(x)=6{x}^{2}-14x-5 \\ f''(x)=12x-14

(i) For concave downwards, f''(x)<0

12x-14<0 \\ 12x<14 \\ x<1\cfrac{1}{6}

(ii) For concave downwards, f''(x)>0

12x-14>0 \\ 12x>14 \\ x>1\cfrac{1}{6}

 

Notice that the point of inflexion does not necessarily have a horizontal gradient. As seen above, the inflexion point is simply where the curve went from being concave downwards to concave upwards. Therefore, it is good practice to check the gradient of tangents before and after the point of inflexion.