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3.02 The second derivative and properties of graphs

Interactive practice questions

Consider the function $f\left(x\right)=x^2-4x+9$f(x)=x24x+9.

a

The leading coefficient of $f\left(x\right)$f(x) is:

Positive

A

Negative

B
b

Hence state the nature of the turning point of this quadratic function.

Maximum

A

Minimum

B
c

Determine the equation for $f'\left(x\right)$f(x).

Enter each line of work as an equation.

d

Determine the equation for $f''\left(x\right)$f(x).

Enter each line of work as an equation.

e

Use your result from part (d) to select the correct statement.

The constant value of $f''\left(x\right)$f(x) indicates that $f\left(x\right)$f(x) is always concave down, indicating a maximum turning point.

A

The constant value of $f''\left(x\right)$f(x) indicates that $f\left(x\right)$f(x) is always concave up, indicating a minimum turning point.

B
Easy
1min

Consider the function $f\left(x\right)=-x^2+4x-9$f(x)=x2+4x9.

Easy
1min

Consider the function $f\left(x\right)=\left(x-5\right)^2+3$f(x)=(x5)2+3.

Easy
3min

Consider the function $f\left(x\right)=-\left(x-5\right)^2-4$f(x)=(x5)24.

Easy
2min
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Outcomes

4.1.1.1

understand the concept of the second derivative as the rate of change of the first derivative function

4.1.1.3

understand the concepts of concavity and points of inflection and their relationship with the second derivative

4.1.1.4

understand and use the second derivative test for finding local maxima and minima

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