Question

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=x^{5}+5 x^{4}-40 x^{2} $$

Short answer

The graph of the function is concave up on the intervals (-4,-1) and (1, ∞), and concave down on the intervals (-∞,-4) and (-1,1).

Step-by-step solution

Differentiate the function

The derivative of the function \(y=x^{5}+5 x^{4}-40 x^{2}\) is given by \[y'=5x^{4}+20x^{3}-80x.\]

Find the second derivative

Differentiate the derivative again to find the second derivative. The second derivative of the function is given by \[y''=20x^{3}+60x^{2}-80.\]

Determine the concavity

Set \(y''=0,\) and solve for \(x\) to find the points at which the concavity might change. These are the potential inflection points. Upon solving we get, \(20x^{3}+60x^{2}-80 =0.\) We simplify to get \(x^{3}+3x^{2}-4=0.\) Solving for \(x\), we have \(x=1\), \(x=-4\), and \(x=-1\). Using these points, we test the intervals \((-∞, -4), (-4,-1), (-1, 1), (1,∞)\) for the sign of \(y''\) to determine where the function is concave up (when \(y''\) is positive) and where it is concave down (when \(y''\) is negative). Testing these intervals, we find that \(y''\) is positive on \((-4,-1)\) and \((1,∞)\), and negative on \((-∞,-4)\) and \((-1,1)\).

Key concepts

Second Derivative

The second derivative of a function provides important information about the concavity of the graph. We start with the original function and find its first and second derivatives. The first derivative tells us the slope or the rate of change of the function. When we differentiate this once more, we obtain the second derivative. For example, in the function given, we differentiated twice to get \( y'' = 20x^3 + 60x^2 - 80 \).

The second derivative plays a crucial role because it indicates whether the curve is bending upwards or downwards. If \( y'' > 0 \) at a particular interval, the graph is concave upward in that interval. Conversely, if \( y'' < 0 \), the graph is concave downward.

Therefore, computing the second derivative is essential in analyzing the shape of the graph.

Inflection Points

Inflection points are where the graph changes concavity. To find these points analytically, we set the second derivative equal to zero, as these are the spots where the concavity transition occurs. From our example, we solved \( 20x^3 + 60x^2 - 80 = 0 \) to get potential inflection points \( x = 1 \), \( x = -4 \), and \( x = -1 \).

To confirm these points are indeed where the concavity changes, further testing is required on the intervals divided by these points. By choosing x-values from these sign-altered intervals and testing them in the second derivative, we verify whether it shifts from positive to negative or vice versa around these points.

• Concave up: \((-4, -1), (1, ∞)\)
• Concave down: \((-∞, -4), (-1, 1)\)

Hence, inflection points help us chart the changes in the graph's direction.

Analytical Techniques

Analytical techniques involve using mathematical strategies to solve calculus problems, like finding concavity and inflection points. It requires understanding derivatives and their implications on the graph's behavior. With the derivative and the second derivative, we have powerful tools to analyze functions.

In our example, we adopted analytical methods to comprehend the given polynomial. Steps included calculating the first derivative to understand slope changes, and the second derivative to determine intervals of concavity. Additionally, solving equations for setting second derivatives to zero gives us inflection points.

Beyond solving, analytical techniques guide us in visualizing connections and changes in behavior, like where the curve dips and rises. Such methods go beyond simple calculations and dive into understanding the intrinsic details of the function being studied.