The function is increasing on the interval calculator

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Calculate the properties of a function step by step

The calculator will try to find the domain, range, x-intercepts, y-intercepts, derivative, integral, asymptotes, intervals of increase and decrease, critical (stationary) points, extrema (minimum and maximum, local, relative, absolute, and global) points, intervals of concavity, inflection points, limit, Taylor polynomial, and graph of the single-variable function. The interval can be specified. Parity will also be determined.

Your input: find the properties of $$$f=x^{3} - 3 x^{2}$$$

Parity

The function is neither even nor odd.

Domain

$$$\left(-\infty, \infty\right)$$$

x-intercepts

$$$\left(0,0\right)$$$

$$$\left(3,0\right)$$$

y-intercepts

$$$\left(0,0\right)$$$

Range

$$$\left(-\infty, \infty\right)$$$

Critical Points

$$$\left(x, f \left(x \right)\right)=\left(0,0\right)$$$

$$$\left(x, f \left(x \right)\right)=\left(2,-4\right)$$$

Intervals of Increase

$$$\left(-\infty, 0\right) \cup \left(2, \infty\right)$$$

Intervals of Decrease

$$$\left(0, 2\right)$$$

Local Minima

$$$\left(x, f \left(x \right)\right)=\left(2,-4\right)$$$

Local Maxima

$$$\left(x, f \left(x \right)\right)=\left(0,0\right)$$$

Global (Absolute) Minima and Maxima

For global minima and maxima, see extrema calculator.

Inflection Points

$$$\left(x, f \left(x \right)\right)=\left(1,-2\right)$$$

Concave upward on

$$$\left(1, \infty\right)$$$

Concave downward on

$$$\left(-\infty, 1\right)$$$

Derivative

For derivative, see derivative calculator.

Integral

For integral, see integral calculator.

Asymptotes

For asymptotes, see asymptote calculator.

Limit

For limit, see limit calculator.

Taylor Polynomial

For Taylor polynomial, see taylor polynomial calculator.

Graph

For graph, see graphing calculator.

Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x)=x^3-75x+3

Step 1

Find the first derivative.

Step 1.1

Find the first derivative.

Step 1.1.1

Step 1.1.1.1

By the Sum Rule, the derivative of with respect to is .

Step 1.1.1.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2

Step 1.1.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2.3

Step 1.1.3

Differentiate using the Constant Rule.

Step 1.1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.1.3.2

Step 1.2

The first derivative of with respect to is .

Step 2

Set the first derivative equal to then solve the equation .

Step 2.1

Set the first derivative equal to .

Step 2.2

Add to both sides of the equation.

Step 2.3

Divide each term in by and simplify.

Step 2.3.1

Step 2.3.2

Step 2.3.2.1

Cancel the common factor of .

Step 2.3.2.1.1

Cancel the common factor.

Step 2.3.2.1.2

Step 2.3.3

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

Step 2.5

Step 2.5.1

Step 2.5.2

Pull terms out from under the radical, assuming positive real numbers.

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

Step 2.6.1

First, use the positive value of the to find the first solution.

Step 2.6.2

Next, use the negative value of the to find the second solution.

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

Step 3

The values which make the derivative equal to are .

Step 4

Split into separate intervals around the values that make the derivative or undefined.

Step 5

Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.

Step 5.1

Replace the variable with in the expression.

Step 5.2

Step 5.2.1

Step 5.2.2

Step 5.2.3

Step 5.3

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Step 6

Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.

Step 6.1

Replace the variable with in the expression.

Step 6.2

Step 6.2.1

Step 6.2.1.1

Raising to any positive power yields .

Step 6.2.1.2

Step 6.2.2

Step 6.2.3

Step 6.3

At the derivative is . Since this is negative, the function is decreasing on .

Decreasing on since

Step 7

Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.

Step 7.1

Replace the variable with in the expression.

Step 7.2

Step 7.2.1

Step 7.2.2

Step 7.2.3

Step 7.3

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on:

Decreasing on:

How do you find the interval of increase?

How do you know when a function is increasing?

Where is a function increasing or decreasing?