RevisionDojo

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

Question 1

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

[5]

Question 2

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 $f(x) = x^3 - 5x^2 + 6x + 9$ .

[2]

Question 3

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

[5]

Question 4

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

[7]

Question 5

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

[4]

Question 6

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

[4]

Question 7

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

[5]

Question 8

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

Determine the intervals on which $f$ is increasing.

[6]

Question 9

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

[4]

Question 10

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

[3]

Question 11

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

[8]

Question 12

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 $f(x) = x^3 - 5x^2 + 6x + 9$ as a local maximum or minimum using the first derivative test.

[4]

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