Question

Determine the open intervals on which the graph is concave upward or concave downward. \(h(x)=x^{5}-5 x+2\)

Short answer

The graph is concave upward on the interval \( (0, +\infty) \) and concave downward on the interval \( (-\infty, 0) \).

Step-by-step solution

Find the first derivative

Differentiate \( h(x) = x^{5} - 5x + 2 \) with respect to \( x \). This yields \( h'(x) = 5x^{4} - 5 \)

Find the second derivative

Differentiate \( h'(x) = 5x^{4} - 5 \) with respect to \( x \). This yields \( h''(x) = 20x^{3} \)

Find intervals where the second derivative is positive or negative

The second derivative \( h''(x) = 20x^{3} \) is positive when \( x > 0 \) and negative when \( x < 0 \).

Identify the intervals of concavity

Since the second derivative \( h''(x) = 20x^{3} \) is positive for \( x > 0 \) and negative for \( x < 0 \), the function \( h(x) = x^{5} - 5x + 2 \) is concave upward on the interval \( (0, +\infty) \) and concave downward on the interval \( (-\infty, 0) \).

Key concepts

Second Derivative Test

When students delve into the study of calculus, understanding the second derivative test is crucial for analyzing the concavity of a function's graph. To ensure that the concept is easily grasped, let's simplify it to its core.

The second derivative test revolves around the calculation of a function's second derivative, denoted as \( h''(x) \). It tells us if a graph is concave upward or downward at certain points. A positive second derivative, \( h''(x) > 0 \), indicates that the graph is concave upward, resembling the shape of a cup that can hold water. Conversely, a negative second derivative, \( h''(x) < 0 \), suggests that the graph is concave downward, like an upside-down cup.

In our given exercise, we found that \( h''(x) = 20x^{3} \), which changes sign depending on the value of \( x \). To identify intervals of concavity, we observe where this second derivative is positive or negative. This approach provides a systematic method for determining the concavity across different intervals of the function's domain.

Concave Upward

When visualizing a graph that is concave upward, I like to imagine a smile. The edges of the graph curve upwards, the same way the corners of a smile lift towards the sky. A graph with this property indicates that the slope of the function is increasing as \( x \) moves from left to right.

In practical terms, a function is concave upward on an interval if its second derivative is positive over that interval. In our example, \( h''(x) = 20x^{3} \) is positive when \( x > 0 \) which tells us that the function \( h(x) = x^{5} - 5x + 2 \) is concave upward for the interval \( (0, +\text{\infty}) \). Simple tasks—like finding where a ball thrown in the air will slow its ascent and begin to fall back down—can be modeled by considering the concavity of a function to identify such turning points.

Concave Downward

On the flip side, a graph that is concave downward can be thought of as a frown, with the edges pointing downwards. In this scenario, the slope of the function decreases as \( x \) moves from left to right, signifying a reduction in the rate of change.

A negative second derivative throughout an interval signifies that the graph of the function is concave downward on that interval. In our exercise, because \( h''(x) = 20x^{3} \) is negative for \( x < 0 \) the graph of \( h(x) = x^{5} - 5x + 2 \) is thus concave downward for the interval \( (-\text{\infty}, 0) \). This knowledge is especially useful in fields like engineering or physics where the direction of curvature can indicate critical stress points or the trajectory of objects under the influence of gravity.