Problem 5 - Calculus of a Single Variable

Question

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=-x^{3}+6 x^{2}-9 x-1 $$

Step-by-Step Solution

Answer

The function \(f(x) = -x^{3}+6x^{2}-9x-1\) is concave upward in the interval (-∞, 2) and concave downward in the interval (2, ∞).

Step 1: Find the First Derivative

The first derivative of the function is obtained using the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). Hence, the first derivative of \(f(x) = -x^{3}+6x^{2}-9x-1\) is \(f'(x) = -3x^2 + 12x - 9\).

Step 2: Find the Second Derivative

The second derivative is the derivative of the first, using the power rule again provides \(f''(x) = -6x + 12\).

Step 3: Find the Intervals

To find where the function is concave upward or downward, set the second derivative equal to zero and solve for x. In this case: \(-6x +12 = 0\) leads to \(x = 2\). It splits the x-axis into two intervals (-∞, 2) and (2, ∞). Check the sign of the second derivative in each interval. Choosing any number \(x_1\) from the interval (-∞, 2), like -1 and evaluating \(f''(x_1)\), gives a positive result, thus this interval is concave upward. Choosing any number \(x_2\) from the interval (2, ∞), like 3 and evaluating \(f''(x_2)\), gives a negative result, meaning this interval is concave downward.

Key Concepts

First Derivative

The first derivative of a function is crucial in calculus as it helps determine the slope of the tangent line to the function at any point. When we take the first derivative of a polynomial like the one in the exercise, we use the power rule. The power rule states that the derivative of a term in the format of \(ax^n\) is \(n \cdot ax^{n-1}\). For this function, \(f(x) = -x^{3}+6x^{2}-9x-1\), applying the power rule gives us the first derivative: \(f'(x) = -3x^2 + 12x - 9\).

• The first derivative tells us the rate of change at any point along the curve.
• It indicates the slope of the tangent, showing where the function is increasing or decreasing.

Understanding the first derivative is essential before we can proceed to explore any concavity of the function. Now that we have the first derivative, we can move on to the second derivative to unravel more about the graph's behavior.

Second Derivative

The second derivative of a function is the derivative of the first derivative and provides valuable information about the curvature of the graph. The second derivative helps us understand the function's concavity, which can tell us where the graph bends upwards or downwards. Using the power rule once more on the first derivative \(f'(x) = -3x^2 + 12x - 9\), we determine the second derivative: \(f''(x) = -6x + 12\).

• The second derivative indicates the rate at which the function's slope is changing.
• It allows us to analyze the concavity of the graph.

By solving \(f''(x) = 0\), we can find the critical point or points that can help determine the intervals of concavity. In this case, setting \(-6x + 12 = 0\) gives \(x = 2\). This point is crucial as it separates regions where the graph is concave upward and downward.

Concave Upward

Concavity describes how a function curves, and a concave upward interval means the graph bends upwards like a cup. For a function to be concave upward on a certain interval, the second derivative must be positive in that interval. To identify these intervals, we examine the sign of the second derivative.In our example, we found \(f''(x) = -6x + 12\) and determined that when \(x < 2\), the second derivative is positive. By picking a test point such as \(x = -1\) and substituting it into \(f''(x)\), we see that the second derivative is positive, hence the interval \((-\infty, 2)\) is concave upward.

• The graph is bowl-shaped in this region, opening up.
• Being able to spot concave upward regions helps in sketching more accurate graphs.

Knowing when the function is concave upward can also inform us about the function's behavior in terms of local maximum and minimum points.

Concave Downward

A concave downward interval occurs when a curve bends downwards, forming a shape like an upside-down cup. This happens in regions where the second derivative is negative. By re-examining our second derivative \(f''(x) = -6x + 12\), we establish that for \(x > 2\), the second derivative is negative. Choosing a test point such as \(x = 3\) shows that \(f''(x)\) is indeed negative, confirming the graph is concave downward in the interval \((2, \infty)\).

• The graph in this region is arch-shaped, opening downwards.
• Understanding concave downward sections is key to identifying possible points of inflection.

Recognizing regions of concave downward helps not only in graphing functions but also in understanding the overall dynamics of the function's change.