Exercises for Question Type 1: Finding intervals where a function is increasing or decreasing

Exercises for Question Type 1: Finding intervals where a function is increasing or decreasing - IB

Written by official IB examiners

Question 1

Skill question

Verify by selecting appropriate test points that $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ is increasing on $(- \infty, \frac{5 - 7}{3})$ and decreasing on $(\frac{5 - 7}{3}, \frac{5 + 7}{3})$ .

[5]

Question 2

Skill question

The question requires finding the first derivative of a cubic polynomial function using the power rule for differentiation.

Find the derivative $f^{'} (x)$ of the function $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ .

[2]

Question 3

Skill question

Identify the critical points of $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ .

[5]

Question 4

Skill question

Determine whether $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ is strictly increasing, strictly decreasing, or neither on the interval $[0, 2]$ .

[7]

Question 5

Skill question

Show that $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ is not monotonic on its entire domain $R$ .

[4]

Question 6

Skill question

Construct a sign chart for $f^{'} (x)$ of $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ , indicating the sign of $f^{'} (x)$ on each interval determined by the critical points.

[4]

Question 7

Skill question

Find the inflection point of $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ by using the second derivative.

[5]

Question 8

Skill question

The function $f$ is defined by $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ for $x \in R$ .

Determine the intervals on which $f$ is increasing.

[6]

Question 9

Skill question

Solve the equation $f^{'} (x) = 0$ for $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ .

[4]

Question 10

Skill question

Determine the intervals on which $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ is decreasing.

[3]

Question 11

Skill question

Using the second derivative test, classify the critical points of $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ .

[8]

Question 12

Skill question

This question assesses the student's ability to apply the first derivative test to classify critical points of a cubic function.

Classify each critical point of $f (x) = x^{3} - 5 x^{2} + 6 x + 9$ as a local maximum or minimum using the first derivative test.

[4]

IB Mathematics Applications & Interpretation Question Type 1: Finding Intervals Where a Function Is Increasing or Decreasing Exercises