Question

Show that Fermat's two methods of determining a maximum or minimum of a polynomial \(p(x)\) are both equivalent to solving \(p^{\prime}(x)=0\)

Short answer

In conclusion, we have demonstrated that both of Fermat's methods for finding the maximum or minimum of a polynomial function, involve finding the points where the first derivative, \(p'(x)\), is equal to zero. The first method looks at the sign change of \(p(x)\), while the second method evaluates the second derivative, \(p''(x)\), at these points to determine whether it is a maximum or minimum. By using either of these methods, we can deduce that finding the points where \(p'(x)=0\) is crucial in determining the extremum of a polynomial function.

Step-by-step solution

Differentiate p(x)

First, we need to find the first derivative of the polynomial function, \(p(x)\). This will give us a function, \(p'(x)\), that represents the slope of \(p(x)\) at any point x.

Analyze the sign change of p(x)

For Fermat's first method, we will analyze the sign change of the function \(p(x)\) for increasing values of \(x\). We need to find when \(p'(x)=0\), as this is the point where there is a local extremum (either a maximum or a minimum). So, we have to find \(x\) such that \(p'(x)=0\).

Find the second derivative of p(x)

Now, let's find the second derivative of the polynomial function, \(p''(x)\). This will give us a function that represents the rate of change of the slope of \(p(x)\) at any point \(x\).

Evaluate the second derivative at the point where the first derivative is zero

For Fermat's second method, we will evaluate the second derivative, \(p''(x)\), at the points where the first derivative is zero (i.e., \(p'(x)=0\)). This will give us insight into whether the extremum is a maximum or a minimum. If \(p''(x)>0\), it implies the function is concave up and there is a local minimum; if \(p''(x)<0\), it implies the function is concave down, and there is a local maximum. By following the steps mentioned above, we have, in both methods, attempted to find the points where \(p'(x)=0\). Thus, Fermat's two methods of determining a maximum or minimum of a polynomial function are both equivalent to solving \(p'(x)=0\).

Key concepts

Derivative of a Polynomial

When analyzing the behavior of a polynomial function, one of the first steps we take is to determine its derivative. The derivative of a polynomial provides us with a new polynomial function that reveals the slope of the tangent line at any point along the original curve. To put it simply, if we have a polynomial function expressed in the form \( p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), the derivative \( p'(x) \) is obtained by applying the power rule to each term. This rule states that for any term \( a_kx^k \), the derivative is \( ka_kx^{k-1} \).

By computing \( p'(x) \) step by step, we can identify the locations on the graph where the slope is zero—these are the potential candidates for local extrema. Understanding the derivative is crucial as it sets the foundation for finding extremum points using Fermat's theorem on stationery points.

Finding Local Extremum

A local extremum of a function \( p(x) \) refers to a point on the graph where the function has either a local minimum or maximum. In other words, near this point, the function doesn't get any smaller (for a minimum) or any larger (for a maximum). Fermat's theorem on stationary points tells us that if a function has a local extremum at a point \( x = c \) and is differentiable there, then \( p'(c) = 0 \).

After finding the derivative \( p'(x) \) of the polynomial, we then solve the equation \( p'(x) = 0 \) to find the potential extrema points. This process involves solving equations, sometimes simple, sometimes complex, depending on the polynomial given. However, not every solution to \( p'(x) = 0 \) will correspond to an actual local extremum—it’s only a potential list of candidates where the extremum can occur.

Second Derivative Test

Once we've identified potential extrema using the first derivative, the second derivative test can help us determine whether these points are indeed local minima or maxima. The second derivative, \( p''(x) \) of our polynomial function gives us information about the curvature of the graph.

If we plug in the values of \( x \) where \( p'(x) = 0 \) into \( p''(x) \), the sign of the result tells us about the nature of the extremum. A positive \( p''(x) \) indicates the graph is curving upwards, so we have a local minimum. Conversely, a negative \( p''(x) \) indicates the graph is curving downwards, pointing to a local maximum. If the second derivative is zero, then the test is inconclusive, and we may need to use higher-order derivatives or other methods to determine the nature of the stationary point.

Solving Equations

In the context of finding the extremum points of a polynomial function, solving equations is the mathematical process used to find the roots of \( p'(x) = 0 \). The roots represent the \( x \) values where the slope of the polynomial function is zero, and potential extrema could occur. Solving these equations can vary in complexity based on the degree of the polynomial.

For linear and quadratic equations, we may use algebraic methods like factoring, completing the square, or applying the quadratic formula. For higher degree polynomials, we might employ synthetic division or numerical methods when analytical solutions are not easily attainable. The essential point is that solving these equations is a step towards identifying where on the graph the function’s behavior changes, setting the stage for maximizing or minimizing the function using calculus.