Question

Multiple Choice If \(a<0,\) the graph of \(y=a x^{3}+3 x^{2}+\) \(4 x+5\) is concave up on (A) \(\left(-\infty,-\frac{1}{a}\right)\) (B) \(\left(-\infty, \frac{1}{a}\right)\) (C) \(\left(-\frac{1}{a}, \infty\right) (D) \)\left(\frac{1}{a}, \infty\right)\( (E) \)(-\infty,-1)$

Short answer

The function is concave up on the interval \( \left( -1/a , \infty \right) \). So, option (C) \(\left(-\frac{1}{a}, \infty\right)\) is the correct answer.

Step-by-step solution

Find the derivative

To begin with, find the derivative of the given function. The derivative of \(y = a x^{3} + 3 x^{2} + 4 x + 5\) is \(y' = 3 a x^{2} + 6 x + 4\).

Find the second derivative

Next, find the second derivative, which is the derivative of the first derivative. The second derivative of y is \( y'' = 6 a x + 6 \).

Set the second derivative equal to zero

To find the points or intervals where the function changes concavity, set the second derivative equal to zero: \( 6 a x + 6 = 0 \) Solving for x, gives \( x = -1/a \).

Determine intervals of concavity

To determine where the function is concave up, evaluate the second derivative at points to the left and right of \( x = -1/a \). Since \( a < 0 \), if you test a number less than \( -1/a \), the second derivative will be negative, hence the function is concave down. And for a number greater than \( -1/a \), the second derivative will be positive meaning the function is concave up. Therefore, the function is concave up on the interval \( \left( -1/a , \infty \right) \).

Key concepts

Second Derivative Test

The second derivative test is a convenient method to determine the concavity of a function. Concavity refers to the direction in which a function curves: upwards like a cup (concave up) or downwards like a frown (concave down).

When applying the second derivative test, you calculate the second derivative of the function and then examine its sign. A positive second derivative indicates that the function is concave up at that point, while a negative second derivative means that the function is concave down. This test is also helpful to locate points of inflection, which occur where the concavity of the function changes—that is, where the second derivative is zero or undefined.

Concave Up and Concave Down Intervals

To understand concave up and concave down intervals, imagine the graph of a function as a road. A concave up interval is where the road curves upwards, forming a valley, such that the slope of the tangent keeps increasing. Conversely, in concave down intervals, the graph forms a hill, and the slope of the tangent decreases.

These concavities are of particular interest because they tell you about the behavior of the function. A function with a graph that's concave up is one where the rate of growth is increasing; meanwhile, a graph that's concave down shows a decreasing rate of growth. Detecting these intervals involves finding the sign of the function’s second derivative over different ranges of the input variable.

Finding Derivatives

Derivatives are the fundamental tool for understanding the behavior of polynomial functions. Simply put, finding the derivative of a function gives you the rate at which the function's output changes with respect to its input. To find the derivative, you apply the rules of differentiation to each term of the function.

Power Rule for Derivatives

• The derivative of a term like \(ax^n\) is found by bringing down the exponent as a coefficient (multiplying it by the coefficient a), and then decreasing the exponent by one.
• For constants (terms without a variable), the derivative is zero, since they do not change as the input changes.

Once you have the first derivative, finding the second derivative involves applying the differentiation rules once more to the result.

Analyzing Function Behavior

Analyzing function behavior goes beyond just plotting the graph. It includes understanding the rate of change (first derivative), the concavity (second derivative), and even higher-order derivatives for more complex behaviors. Critical points, inflection points, increasing and decreasing intervals, and the end behavior of the function are all aspects to consider when analyzing.

By combining the information from both the first and second derivatives, you can start to paint a comprehensive picture of the function’s behavior across its domain. This includes where it's rising or falling, where it's speeding up or slowing down, and where significant changes in curvature occur. Functions also have practical implications as they can model various real-world scenarios, so analyzing their behavior is not just a mathematical exercise but also an applicable skill in various sciences.