Problem 60 - Algebra and Trigonometry

Question

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{3}+12 x $$

Step-by-Step Solution

Answer

The polynomial has no local maxima or minima.

Step 1: Find the First Derivative

To determine the points where local maxima and minima may occur, we first find the first derivative of the polynomial. Given the function \( y = x^3 + 12x \), the first derivative is \( y' = \frac{d}{dx}(x^3 + 12x) \). Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \), we find: \[ y' = 3x^2 + 12. \]

Step 2: Solve for Critical Points

Set the first derivative equal to zero to find the critical points, as any local maxima or minima must occur where the derivative is zero. Solve \( 3x^2 + 12 = 0 \): \[ 3x^2 + 12 = 0 \]\[ 3x^2 = -12 \]\[ x^2 = -4. \] However, \( x^2 = -4 \) has no real solutions as the square of a real number cannot be negative. Thus, no real critical points exist in this function.

Step 3: Graph the Function

Even though there are no real critical points, we graph the function \( y = x^3 + 12x \) to visualize its behavior. Notice that the function is a cubic polynomial which generally does not have local maxima and minima without real critical points. The graph shows that the function is increasing everywhere with no peaks or valleys.

Step 4: Confirm Interpretation

With no real critical points from the derivative and observing the cubic polynomial graph, it confirms the understanding that there are no local maxima or minima. Conclude from both algebraic and graphical evidence.

Key Concepts

Derivatives

The concept of derivatives plays a crucial role in calculus and helps us understand how a function behaves. For any given function, the derivative represents the rate at which the function’s value changes as its input changes. In simpler terms, it tells us the slope of the function at any point. This is essential in determining where a function increases or decreases.

To find the derivative of a polynomial function like \( y = x^3 + 12x \), we apply the power rule. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Using this rule, we find that the derivative of the given function is \( y' = 3x^2 + 12 \). This derivative can now be used to identify critical points, potential peaks, or valleys on the graph.

Critical Points

Critical points are where the function’s derivative equals zero or is undefined. It's important because these points can potentially be where the function changes direction, leading to local maxima or minima. To find critical points, you set the derivative to zero and solve for \( x \).

In our example, to find the critical points of \( y = x^3 + 12x \), we solve \( 3x^2 + 12 = 0 \). However, we encounter a problem: the equation \( x^2 = -4 \) has no real solutions. This tells us that there are no points where the slope is zero, meaning there are no real critical points in this function.

Local Maxima and Minima

Local maxima and minima refer to the highest and lowest points in a particular region of a function's graph. These are places where the function stops increasing and starts decreasing, or vice versa.

For the polynomial \( y = x^3 + 12x \), since there are no real critical points, it indicates that there are no local maxima or minima. This function does not have any peaks or valleys within the real number system, as confirmed by both algebraic methods and graphical representation.

Graphing Polynomials

Graphing polynomials can provide visual insights into the behavior of functions. A polynomial's graph can help us see where a function increases or decreases and identify any apparent maxima or minima.

For our function \( y = x^3 + 12x \), graphing it confirms that the function is continuously increasing. With its cubic nature, the graph displays a smooth curve with no real peaks or valleys. Although it has no local maxima or minima, understanding how to interpret these graphs is key in revealing critical insights into different functions. This comprehensive view helps us confirm theoretical findings with visual representation.