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# Concavity Calculator Online

## Introduction

Concavity is a fundamental concept in calculus that helps to understand the shape of a graph. It reveals whether a function's slope is increasing or decreasing. A concavity calculator simplifies the process of finding where a function is concave up or concave down. This article walks you through the workings of a concavity calculator, along with the underlying formula and a practical example.

## What is a Concavity Calculator?

A concavity calculator is an online tool used to determine the nature of a function—whether it's concave up, concave down, or experiencing an inflection point at a given interval. The calculator uses the principles of the second derivative test in calculus to make this determination.

## Working of a Concavity Calculator

The concavity calculator works on the basis of the second derivative test. The key steps are as follows:

1. The user enters the function and the specific x-value.
2. The calculator evaluates the second derivative of the function at this x-value.
3. The concavity of the function at this point is determined based on the result:
• If the second derivative is positive, the function is concave up.
• If the second derivative is negative, the function is concave down.
• If the second derivative equals zero, the function has an inflection point.

## The Underlying Formula

The general approach to determine the concavity involves finding the second derivative of the function, expressed as f''(x). The concavity is then determined as follows:

• f''(x) > 0, the function is concave up.
• f''(x) < 0, the function is concave down.
• f''(x) = 0, the function has an inflection point.