Exploring Second derivatives

Here is an example of a curve y that concaves up. Its derivative dy/dx is ____.

Hint: constant, positive, negative, zero.

The function y here concaves down.

Its derivative y'(x) is ____.

Hint: zero, constant, positive, negative.

Answer explanation

The function is still increasing and thus y' > 0. So y' does not tell us anything about concavity.

Let y = (x+1)³. Find dy/dx.

• 3(x+1)

• 3(x+1)²

• (x+1)²

For y =(x+1)^3, its derivative is always positive.

Use a geometric approach to explain

why dy/dx is always positive.

The second derivative is the derivative of the y' or dy/dx.

The second derivative is written as y'' or d²y/dx² (d two y by d x squared).

These statements are ____.

• True.

• False.

Let y = (x-1)³ then y' =3(x-1)².

Find y''.

• 3(x-1)

• 6(x-1)

• -6(x-1)

Let y = (x-1)³ then y'' =6(x-1).

Find x when y" > 0.

What is shape of the function when y" > 0?

Hint: GDC.

• x > 1

  and

  y concaves up

• x > 1

  and

  y concaves down

• x < 1

  and

  y concaves up

• x < 1

  and

  y concaves down

Let y = (x-1)³ then y'' =6(x-1).

Find x when y" < 0.

What is shape of the function when y" < 0?

Hint: GDC.

• x > 1

  and

  y concaves up

• x > 1

  and

  y concaves down

• x < 1

  and

  y concaves up

• x < 1

  and

  y concaves down

Let y = (x-2)³ then y'' =6(x-2).

Find x when y" = 0.

What is the significant of this point in terms of concavity of this curve?

Hint: GDC.

• x = 0

  Concavity changes at this point.

• x = 2

  Concavity changes at this point.

If (x₀,y₀) is an inflection point then y"(x₀) is ____.