Math Problem Statement

The open intervals on which f is increasing. (Enter your answer using interval notation.)

The open intervals on which f is increasing.

Solution

To determine the open intervals on which the function f(x)f(x) is increasing, we need to identify the regions where the slope of the curve is positive (i.e., where the graph is moving upward from left to right).

From the graph provided, the function increases in the following intervals:

  1. From x=x = -\infty up to the first turning point, which is roughly at x0.3x \approx 0.3.
  2. From the next local minimum, around x1.7x \approx 1.7, until the graph continues to rise indefinitely.

Thus, the function f(x)f(x) is increasing on the following intervals:

(,0.3)(1.7,)(-\infty, 0.3) \cup (1.7, \infty)

Would you like further explanation, or do you have any questions?

Here are five related questions you might find helpful:

  1. How can you find the intervals where a function is decreasing?
  2. What is the significance of local minima and maxima for identifying intervals of increase or decrease?
  3. How do the first and second derivatives relate to the intervals where a function is increasing?
  4. Can a function be increasing or decreasing at a single point, like a critical point?
  5. What does it mean when a function is constant over an interval?

Tip: Remember, a function is increasing when the derivative f(x)>0f'(x) > 0 and decreasing when f(x)<0f'(x) < 0.

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Math Problem Analysis

Mathematical Concepts

Calculus
Graph Analysis

Formulas

First Derivative Test

Theorems

Critical Points
Increasing/Decreasing Test

Suitable Grade Level

Grades 10-12