Problem 57 - Calculus Single Variable

Question

Plot \(f(x)=12 x^{5}-2565 x^{4}+146200 x^{3}+1\) for \(x \in I=\) 0,300]\(.\) Plot \(f^{\prime}\) and \(f^{\prime \prime}\) for \(x \in I\). Is \(f\) increasing on \(I ?\) Is \(f^{\prime}>0\) on \(I\) ? Is the graph of \(f\) concave up on \(I\) ? Is \(f^{\prime \prime}>0\) on \(I\) ?

Step-by-Step Solution

Answer

Check the plots to verify if \(f' > 0\) for increasing function and if \(f'' > 0\) for concavity up on \([0, 300]\).

Step 1: Understand the Problem

We need to analyze the function \(f(x) = 12x^5 - 2565x^4 + 146200x^3 + 1\) over the interval \([0, 300]\). The task involves plotting \(f\), its first derivative \(f'\), and its second derivative \(f''\). We then need to determine whether \(f\) is increasing, whether \(f' > 0\) throughout the interval, if \(f\) is concave up, and whether \(f'' > 0\) on the interval \(I\).

Step 2: Calculate First Derivative

To find \(f'\), differentiate \(f(x) = 12x^5 - 2565x^4 + 146200x^3 + 1\) with respect to \(x\). This gives us: \[f'(x) = 60x^4 - 10260x^3 + 438600x^2.\]

Step 3: Calculate Second Derivative

Differentiate \(f'(x) = 60x^4 - 10260x^3 + 438600x^2\) to find \(f''(x)\). This gives: \[f''(x) = 240x^3 - 30780x^2 + 877200x.\]

Step 4: Plot Function and Its Derivatives

Using a graphing tool, plot \(f(x) = 12x^5 - 2565x^4 + 146200x^3 + 1\), \(f'(x) = 60x^4 - 10260x^3 + 438600x^2\), and \(f''(x) = 240x^3 - 30780x^2 + 877200x\) over the interval \([0, 300]\).

Step 5: Analyze the Plots

Examine the plot of \(f(x)\) to determine if it is increasing, by checking if the slope (\(f'\)) is always positive. Next, examine \(f'(x)\) to see if \(f' > 0\) over \([0, 300]\). Lastly, use the plot of \(f''(x)\) to check for concavity (concave up if \(f'' > 0\)).

Step 6: Conclusion

From the plots: If \(f'(x) > 0\) over the entire interval, then \(f\) is increasing on \([0, 300]\). Check if \(f''(x) > 0\) to determine if \(f\) is concave up over \([0, 300]\).

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus as it measures how a function changes as its input changes. In simpler terms, it tells us the rate at which a function is increasing or decreasing at any given point. For our function, \(f(x) = 12x^5 - 2565x^4 + 146200x^3 + 1\), the first derivative, \(f'(x)\), is \(60x^4 - 10260x^3 + 438600x^2\).

• The derivative gives us a new function that represents the slope or the steepness of the original function \(f(x)\).
• In our context, finding \(f'(x)\) helps us understand whether our function \(f\) is increasing or decreasing over the interval \([0, 300]\).

By examining where \(f'(x)\) is positive, negative, or zero, we can determine the nature of the function \(f(x)\) across this range.

Function Analysis

Function analysis involves understanding the behavior of a function by examining its derivatives. The first derivative provides insight into the slope of the function, while the second derivative informs us about the curvature of the function.

• When analyzing a function like \(f(x)\), we first compute its derivative \(f'(x)\), which shows where the function is increasing or decreasing.
• Next, by analyzing the second derivative \(f''(x)\), we can grasp more about the behavior of \(f\), particularly its concavity.

Through this analysis, we draw conclusions on whether the function has peaks (local maxima), valleys (local minima), or points of inflection. This understanding is critical in various fields that require optimization or modeling of data-driven scenarios.

Concavity

Concavity describes the curvature or the shape of the graphed function, specifically whether it "cups" upwards or downwards. This is determined by the sign of the second derivative \(f''(x)\).

• If \(f''(x) > 0\), the graph of \(f\) is concave up, meaning it resembles a cup opening upwards.
• If \(f''(x) < 0\), the graph is concave down, like an upside-down bowl.

For our function, the second derivative \(f''(x) = 240x^3 - 30780x^2 + 877200x\) gives crucial insights into its shape across the interval \([0, 300]\). This aspect of function analysis helps in understanding not just the direction of increase or decrease but the manner in which the function progresses.

Increasing Function

An increasing function is one that consistently moves upwards as you move from left to right along the graph. To determine if a function is increasing on an interval, we look at the sign of the first derivative \(f'(x)\).

• When \(f'(x) > 0\), the function is increasing.
• If \(f'(x) < 0\), the function is decreasing.

In our analysis, we examine \(f'(x) = 60x^4 - 10260x^3 + 438600x^2\) over the interval \([0, 300]\). By checking the values at different points, we determine where \(f\) actually increases. This concept of determining increasing or decreasing sections is essential in graphing functions and real-world applications such as economics and natural sciences where trends and forecasts are crucial.