Question

Use a graphing calculator or a computer to do. Let \(f^{\prime \prime}(x)=x^{4}-5 x^{3}+4 x^{2}+4\) on \(I=[-2,3]\). Where on \(I\) is \(f\) concave down?

Short answer

Function \(f\) is concave down where \(f''(x) < 0\) on \([-2, 3]\).

Step-by-step solution

Understanding Concavity

A function is concave down on an interval if its second derivative is negative on that interval. Here, we are given the second derivative \(f''(x) = x^4 - 5x^3 + 4x^2 + 4\) and need to determine where it is negative over the interval \([-2, 3]\).

Graphing the Second Derivative

Use a graphing calculator or computer software to plot the function \(f''(x) = x^4 - 5x^3 + 4x^2 + 4\) on the interval \([-2, 3]\).

Key concepts

Concavity and its Importance

The concept of concavity provides important information about the behavior of a function. If a function is concave down on an interval, this means the curve of the function is bending downwards like a frown. Concavity can be determined by analyzing the second derivative of the function. A key insight here is that if the second derivative is negative over an interval, then the function is concave down in that region. Recognizing intervals of concavity helps us understand the overall shape and behavior of the graph.

Understanding the Second Derivative

The second derivative, denoted by \(f''(x)\), is a derivative of the derivative. It tells us about the rate of change of the slope of the original function \(f(x)\). Here's why it matters:

• When \(f''(x) > 0\), the function is concave up, indicating that the slope of \(f(x)\) is increasing.
• Conversely, when \(f''(x) < 0\), the function is concave down, meaning the slope of \(f(x)\) is decreasing.
• If \(f''(x) = 0\), the graph could be transitioning between concave up and concave down, often indicating a point of inflection.

By understanding these key points, we can better predict the behavior and transition points of functions. In the given problem, we analyze \(f''(x) = x^4 - 5x^3 + 4x^2 + 4\) to find where the function is concave down.

Using a Graphing Calculator

Graphing calculators or computer software like Desmos or GeoGebra are invaluable tools in calculus for visualizing functions and their derivatives. Here's why they are helpful:

• They provide a quick visual representation of complex functions, where plotting by hand would be tedious or infeasible.
• Different characteristics of the function, like points of inflection or regions of concavity, become immediately visible.
• This visual aid helps confirm mathematical calculations when plotted over specified intervals like \([-2, 3]\).

Using the software, you can enter the function \(f''(x) = x^4 - 5x^3 + 4x^2 + 4\) and observe where the graph dips below the x-axis (indicating concavity down) between \(x = -2\) and \(x = 3\). This greatly simplifies the analysis involved.

Function Analysis and its Applications

Function analysis combines both visual and algebraic methods to deeply understand a function's behavior. Here's why it's crucial:

• By analyzing the derivatives, we gain insight into the increasing or decreasing nature, as well as the concavity of functions.
• It allows us to determine critical points, which are where changes in behavior occur, such as peaks, troughs, and inflection points.
• Function analysis is not just limited to simple functions; it can extend to real-world problems like physics, economics, and engineering to forecast trends and changes.

In our exercise, analyzing \(f''(x)\) through graphs and mathematical techniques allows us not only to determine regions of concavity but also provides a comprehensive understanding of how the function behaves across different intervals.