Standard Level

11.12 Graphs and the second derivative

Lesson

Worksheet

Practice

Worksheet

Higher derivatives

Suppose y = \dfrac{2}{x^{5}}. Find:

\dfrac{d y}{d x}

\dfrac{d^{2} y}{d x^{2}}

\dfrac{d^{3} y}{d x^{3}}

Suppose y = 5 x^{8}. Find:

\dfrac{d y}{d x}

\dfrac{d^{2} y}{d x^{2}}

\dfrac{d^{3} y}{d x^{3}}

\dfrac{d^{4} y}{d x^{4}}

\dfrac{d^{5} y}{d x^{5}}

Consider the function y = - 2 x^{4} + 4 x^{3} - 7 x^{2} + 8 x - 9. Find:

\dfrac{d y}{d x}

The value of \dfrac{d y}{d x} when x = 3

\dfrac{d^{2} y}{d x^{2}}

The value of \dfrac{d^{2} y}{d x^{2}} when x = 4

\dfrac{d^{3} y}{d x^{3}}

The value of \dfrac{d^{3} y}{d x^{3}} when x = 6

Find the second derivative of the following functions:

y = x^4 - 6x^3 + 8x^2 - 10

f \left( x \right) = x^{\frac{1}{7}}

f \left( x \right) = \left( 3 x + 2\right)^{4}

f \left( x \right) = \dfrac{4}{\left(x + 8\right)^{2}}

y = \left(7 - 6 x^{2}\right)^{\frac{1}{3}}

y = 4 \sqrt{6 - x}

f \left( x \right) = x^{2} \left(x^{3} + 3\right)

f \left( x \right) = \left(x^{4} + 2\right) \left(3 - 2 x^{3}\right)

Concavity

Consider the given function y = f(x) with points A, B, C and D:

At which of these points is f''(x) > 0 ?

At which of these points is f''(x) <0 ?

For each of the following curves, state the sign of f'(a) and f''(a):

For each of the following quadratic functions:

State whether the leading coefficient of f \left( x \right) is positive or negative.

Hence, determine the nature of the turning point.

iii

Find f' \left( x \right).

Find f'' \left( x \right).

State whether the curve is concave up or down.

f \left( x \right) = x^{2} - 4 x + 9

f \left( x \right) = - x^{2} + 4 x - 9

Consider the function f \left( x \right) = 4 x^{2} + 3 x + 2.

Find the first derivative.

Find the second derivative.

Complete the following tables of values:

x	0	1	2	3	4
f' \left( x \right)	3

Describe the rate of change of the gradient.

Are there any points of inflection on f(x)? Explain your answer.

Consider the function f \left( x \right) = -3 x^{2} + 12 x + 2.

Find f' \left( x \right).

Find f'' \left( x \right).

Describe the rate of change of the gradient.

Are there any points of inflection on f(x)? Explain your answer.

For each of the following functions:

State the values of x for which the graph of the function is concave up.

State the values of x for which the graph of the function is concave down.

iii

Explain why there is no point of inflection.

y=\sqrt{x+1}

y=-\sqrt{x-2}

y=\dfrac{1}{x}

y=\dfrac{1}{x^2}

Consider the function y = \left(x + 6\right)^{3}.

State the transformation that turns y = x^{3} into y = \left(x + 6\right)^{3}.

Find the point of inflection of y = x^{3}.

Find the point of inflection of y = \left(x + 6\right)^{3}.

Complete the following table of values:

Is the point of inflection, a horizontal point of inflection? Explain your answer.

For what values of x is the graph concave up?

x	- 7	- 6	- 5
y'
y''		0

Consider the function y = \left(x + 5\right)^{3}.

State the transformation that turns y = x^{3} into y = \left(x + 5\right)^{3}.

Find the point of inflection of y = x^{3}

Find the point of inflection of y = \left(x + 5\right)^{3}.

Complete the following table of values:

Is the point of inflection, a horizontal point of inflection? Explain your answer.

For what values of x is the graph concave up?

x	- 6	- 5	- 4
y'
y''		0

Consider the function y = x^{4} - 8 x^{3} - 9.

Find y''.

Find the points of inflection.

Complete the following table of values:

Classify each point of inflection as an ordinary or horizontal point of inflection.

For what values of x is the graph concave up?

x	- 2	0	2	4	6
y'
y''		0		0

Consider the function y = 4 x^{3} - 16 x^{2} + 4 x + 6.

Find y''.

Find the point of inflection.

Is the point of inflection, a horizontal or an ordinary point of inflection?

For what values of x is the graph concave down?

Consider the function y = 3 x^{3} + 6 x^{2} + 8 x + 3.

Find y''.

Find the point of inflection.

Is the point of inflection, an ordinary or horizontal point of inflection?

For what values of x is the graph concave down?

Consider the function y = x^{5} - 3 x^{2}.

Find y''.

Find the exact point of inflection.

Is the point of inflection an ordinary or a horizontal point of inflection?

For what values of x is the graph concave up?

For what values of x is the graph concave down?

The first derivative of a certain function is f' \left( x \right) = 3 x^{2} + 9 x.

Determine the interval over which the function is increasing.

Determine the interval over which the function is decreasing.

Find f'' \left( x \right).

Determine the interval over which the function is concave up.

Determine the interval over which the function is concave down.

Find the x-coordinate of the maximum turning point.

Find the x-coordinate of the potential point of inflection.

Stationary points

Consider the function f \left( x \right) = \left(x - 5\right)^{2} + 3.

Find the x-coordinate of the turning point.

Find the value of the second derivative at this point.

Hence, classify the stationary point.

Consider the function f(x)=(x-5)(x+4)^2.

Find the turning points.

Find the second derivative.

Determine the nature of the turning points.

State the values of x where the graph of f(x) is concave down.

Consider the function y = x^{4} + 4 x^{3} + 2.

Find y'.

Find y''.

Find the stationary points.

Classify the stationary points.

Find the points of inflection.

Classify each point of inflection as ordinary or horizontal point of inflection.

For what values of x is the graph concave up?

Consider the function f \left( x \right) = \left(x - 8\right) \left(x - 5\right)^{2}.

Find f' \left( x \right).

Find the turning points.

Find f'' \left( x \right).

Classify the turning points.

For what values of x is the graph concave down?

For each of the following functions:

Find the turning points.

Classify the turning points.

f(x)=x^3-27x-7

f(x)=x^3-12x-2

f(x)=2x^3+9x^2-24x

f(x)=x^4-2x^2+3

For each of the following functions:

Find the x-coordinates of the stationary points.

Determine the nature of the stationary points.

f \left( x \right) = 7 x^{3} + 8 x^{2} + 3 x + 4

f \left( x \right) = 8 \left(x - 7\right)^{2} \left(x + 2\right)

f \left( x \right) = - \dfrac{8}{x^{2}} + 2 x

f \left( x \right) = 2 x^{3} - 30 x^{2} + 150 x - 247

f \left( x \right) = 2 \left(x - 5\right)^{3} + 1

Consider the function f \left( x \right) = 5 x^{3} + 7 x^{2} + 3 x + 6.

Find the x-coordinates of stationary points.

Find the x-value of the point of inflection.

How many stationary points are there?

Consider the function y = \left(x^{2} - 5\right)^{3}.

Find y''.

Find the points of inflection.

Which of these points are horizontal points of inflection?

For each of the following functions:

Find the stationary point.

Find the possible point of inflection.

iii

Determine whether this point is a turning point or a point of inflection. Explain your answer.

y = 4 - \left(x - 5\right)^{4}

y = 5 - \left(x - 4\right)^{4}

y = 6 - \left(x - 8\right)^{4}

The function f \left( x \right) = a x^{2} + \dfrac{b}{x^{2}} has turning points at x = 1 and x = - 1.

Use the fact that there is a turning point at x = 1 to form an equation for a in terms of b.

Use the fact that there is a turning point at x = - 1 to form an equation for a in terms of b.

What can you deduce about the values of a and b?

The function f \left( x \right) = a x^{3} + b x^{2} + 12 x + 5 has a horizontal point of inflection at x = 1.

Use information about f' \left( x \right) to write an equation involving a and b.

Use information about f'' \left( x \right) to write an equation for b in terms of a.

Hence, find the value of a.

Hence, find b.

The function f \left( x \right) = a x^{3} + 18 x^{2} + c x + 4 has turning points at x = 4 and x = 2. Find the value of:

11.12 Graphs and the second derivative

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