Question

Justify Fermat's first method of determining maxima and minima by showing that if \(M\) is a maximum of \(p(x)\), then the polynomial \(p(x)-M\) always has a factor \((x-a)^{2}\), where \(a\) is the value of \(x\) giving the maximum.

Short answer

Question: Justify Fermat's first method of determining maxima and minima for a polynomial function which states that if M is a maximum value of the polynomial p(x), then the polynomial p(x) - M will have a factor of the form (x-a)^2, where 'a' is the value of x at which the maximum occurs. Answer: Fermat's first method can be justified using the properties of derivatives and the definition of maximum for a polynomial function. By calculating the first and second derivatives of the polynomial p(x), finding the critical points where the first derivative is zero, and checking if the second derivative is negative at those points, we can identify the maximum value of the function at x=a. Using the Taylor series expansion, we showed that if there is a maximum at x=a, the equation p(x) - M will always have a factor of (x-a)^2.

Step-by-step solution

Calculate the first and second derivatives of polynomial p(x)

To find the maximum value of a polynomial function, we need to analyze the function's behavior at different points. One method is to use the first and second derivatives of the function. Let p'(x) be the first derivative of p(x) and p''(x) be the second derivative of p(x). Compute p'(x) and p''(x) for a given polynomial p(x).

Find the critical points of the polynomial function

To find the maximum value of p(x), we need to identify where the function changes its behavior. These points are called critical points and occur where the first derivative is zero or does not exist. Use p'(x) to find the critical points by solving the equation p'(x) = 0.

Check if given critical point corresponds to a maximum

Now we have the critical points from step 2, and we want to check if at the given \(x=a\) we have a maximum. The second derivative test can be used to determine if a critical point represents a maximum, minimum, or saddle point. If p''(a) < 0 at a critical point x=a, then p(x) has a maximum value at x=a.

Prove that p(x) - M has a factor (x-a)^2

We are given that M is the maximum value of p(x), and we wish to prove that p(x) - M has a factor (x-a)^2. Since there is a maximum at x=a, we know that p'(a) = 0 and p''(a) < 0. Now let's consider the Taylor series expansion of p(x) around x=a: \(p(x) = p(a) + p'(a)(x-a) + \frac{1}{2}p''(a)(x-a)^2 + \cdots\) Since p'(a) = 0, the equation reduces to: \(p(x) = p(a) + \frac{1}{2}p''(a)(x-a)^2 + \cdots\) As we are now considering the maximum value M, we can replace p(a) by M: \(M = p(a) = p(x) - \frac{1}{2}p''(a)(x-a)^2 - \cdots\) Therefore, \(p(x) - M = \frac{1}{2}p''(a)(x-a)^2 + \cdots\) Since p''(a) < 0, we note that the polynomial will always have a term (x-a)^2 with a leading coefficient (which is \(\frac{1}{2}p''(a)\) in this case). It implies p(x)-M always has a factor of (x-a)^2. Thus, we have justified Fermat's first method for determining maxima and minima.

Key concepts

Maxima and Minima

In calculus, finding the maxima and minima of a function helps determine the highest or lowest points on a graph. These points are crucial for understanding the behavior of a polynomial function. A maximum point refers to the peak of a curve, while a minimum represents the lowest dip.
When a function has a maximum at a point, it means the function value at that point is greater than or equal to the value of the function at any nearby points. Similarly, a minimum means the function value at that point is the smallest around.
Maxima and minima are found at critical points, where the behavior of the function might change from increasing to decreasing or vice versa.

Derivative Test

Derivatives are essential tools in calculus used to find critical points and determine if these points are maxima, minima, or saddle points. The derivative test involves both the first and second derivatives of a function.
The first derivative of a function, denoted as \( p'(x) \), helps locate critical points, where the slope of the tangent is zero or undefined. Solving \( p'(x) = 0 \) gives potential points where maxima or minima may occur.

• If the first derivative test shows \( p'(a) = 0 \), but the second derivative \( p''(a) \) is negative, it indicates a maximum at \( x = a \).
• If \( p''(a) \) is positive, it suggests a minimum at \( x = a \).
• If \( p''(a) = 0 \), further testing or analysis may be needed.

This derivative test is a straightforward method for analyzing the nature of critical points.

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of variables with coefficients. These functions are represented as \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \.\.\.+ a_1x + a_0 \), where \( a_n, a_{n-1}, \.\.\., a_0 \) are constants.
Understanding polynomial functions is key in calculus as they often form the basis of modeling natural phenomena. They can be simple linear equations or complex higher-degree equations.

• Linear polynomials have degree 1 with the form \( ax + b \).
• Quadratic polynomials have degree 2, like \( ax^2 + bx + c \).
• Cubic polynomials have degree 3, \( ax^3 + bx^2 + cx + d \).

Higher degree polynomials can have multiple maxima or minima, making the use of derivatives essential in analyzing their behavior. Through calculus, especially by applying derivative tests, we gain insights into the nature and movement of these polynomial functions.