Problem 1 - Calculus with Concepts in Calculus

Question

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=-\frac{3}{2} x^{2}+x $$

Step-by-Step Solution

Answer

The function is concave downward everywhere since the second derivative is negative.

Step 1: Determine the second derivative

The concavity of a function is determined by its second derivative. First, we find the first derivative of the function:\[ f(x) = -\frac{3}{2}x^2 + x \]The first derivative is:\[ f'(x) = -3x + 1 \]Now, we find the second derivative by differentiating \( f'(x) \):\[ f''(x) = -3 \]

Step 2: Analyze the second derivative

The second derivative \( f''(x) = -3 \) is a constant. This means it does not change with respect to \( x \). A negative constant indicates that the function is concave downward across its entire domain.

Key Concepts

Second Derivative Test

The Second Derivative Test is a powerful tool in calculus which helps determine the concavity of a function and even identify potential points of inflection. To begin with, we must find the second derivative of the function in question. This requires two steps: finding the first derivative, and then differentiating it to obtain the second derivative.

To quickly recap, if a function is given by \( f(x) = -\frac{3}{2}x^2 + x \), its first derivative, representing the rate of change of the function, is \( f'(x) = -3x + 1 \). Differentiating this again gives the second derivative \( f''(x) = -3 \).

The second derivative helps us understand the behavior of the graph. If \( f''(x) > 0 \) for all \( x \) in an interval, the graph is concave upward on that interval. If \( f''(x) < 0 \), then the graph is concave downward on that interval. This test makes it simple to classify the curvature of the graph into those categories.

Concave Upward

A function is said to be concave upward on an interval where the curve of the graph resembles a cup facing upwards. The defining property of such an interval is that the second derivative is positive: \( f''(x) > 0 \).

In this scenario, if you imagine filling the section of the curve with water, it would be able to hold the water, thanks to its upward cup-like shape. Concavity indicates the behavior of the function in terms of acceleration or "curvature" over a specific range of \( x \).

However, in our exercise, the second derivative, \( f''(x) = -3 \), is constant and negative. Hence, there are no sections where the graph is concave upward. This is important as it implies no positive curvature exists for this function across its domain.

Concave Downward

When a function is concave downward on an interval, its graph looks like a frown or a cup facing downwards. Here, the crucial point is that the second derivative is negative: \( f''(x) < 0 \).

This results in the graph having a downward bend, and metaphorically, it would "spill" water if you tried to fill such a curve. The negative value of the second derivative indicates that the slope of the function's tangent decreases over the interval.

In our original problem, the second derivative \( f''(x) = -3 \) is consistently negative. Thus, the entire graph of this function is concave downward across all values of \( x \). This uniformity in the function’s curvature simplifies the task of analyzing the graph's shape, making it constantly concave down throughout its domain.