Question

Consider the general cubic polynomial \(f(x)=x^{3}+a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are real numbers. a. Prove that \(f\) has exactly one local maximum and one local minimum provided \(a^{2}>3 b\) b. Prove that \(f\) has no extreme values if \(a^{2}<3 b\)

Short answer

Answer: If \(a^2 > 3b\), \(f(x)\) will have exactly one local maximum and one local minimum. If \(a^2 < 3b\), \(f(x)\) will have no extreme values.

Step-by-step solution

Find the derivative of \(f(x)\)

To find the critical points on \(f(x)\), we first need to find its derivative. The derivative is given by: \(f'(x) = \frac{d}{dx}(x^3 + ax^2 + bx + c) = 3x^2 + 2ax + b\)

Determine the critical points

Critical points are values of \(x\) that make the derivative equal to zero or undefined. In our case, the derivative is always defined, so we can focus on finding the values of \(x\) for which \(f'(x) = 0\): \(3x^2 + 2ax + b = 0\) To further understand the behavior of the function around its critical points, we will have to find the second derivative of \(f(x)\).

Find the second derivative of \(f(x)\)

The second derivative will be the derivative of the first derivative, \(f'(x)\). It is given by: \(f''(x) = \frac{d}{dx}(3x^2 + 2ax + b) = 6x + 2a\) Now, let's proceed to analyze the conditions specified in the exercise and their implications for the extreme values of the function.

Prove if \(a^2 > 3b\), \(f(x)\) has exactly one local maximum and one local minimum

First, we will solve the first derivative for x: \(x = \frac{-2a \pm \sqrt{(-2a)^2 - 4(3)(b)}}{2(3)}\). Now, if \(a^2 > 3b\), it follows that the discriminant is positive, which means we have two distinct real roots. Given f'(x) has degree 2 and has two distinct real roots, then f(x) must have at least one local maximum and one local minimum. Now, let's determine the behavior of the function in these critical points using its second derivative. For points \(x_1\) and \(x_2\): - If \(f''(x) > 0\), the function is concave up, indicating a local minimum. - If \(f''(x) < 0\), the function is concave down, indicating a local maximum. Since the second derivative is a linear function, it follows that if one of these points is greater than zero, the other must be smaller than zero, and therefore, there is exactly one local maximum and one local minimum.

Prove if \(a^2 < 3b\), \(f(x)\) has no extreme values

If \(a^2 < 3b\), it follows that the discriminant is negative, meaning we won't have any real roots for the equation \(f'(x) = 0\). As a consequence, the first derivative has no real zeros, so there can be no local maxima or minima. Consequently, \(f(x)\) has no extreme values. In conclusion: a. If \(a^2 > 3b\), \(f(x)\) has exactly one local maximum and one local minimum. b. If \(a^2 < 3b\), \(f(x)\) has no extreme values.

Key concepts

Critical Points in Calculus

In calculus, critical points are essential for understanding the behavior of functions. They are the x-values where the derivative of the function is either zero or undefined. These points are helpful to analyze as they can indicate where a function's graph has a horizontal tangent line and are potential locations for local maxima and minima, which are the highest or lowest points in a given interval, respectively.

For the cubic polynomial mentioned, the critical points are obtained by setting the first derivative equal to zero, providing us with an equation to solve. Understanding critical points gives us valuable insight into the function's inflection points and intervals of increase or decrease.

Discriminant of a Polynomial

The discriminant of a polynomial is a value that can be calculated from the coefficients of the polynomial, and it provides critical information about the roots of the polynomial without actually solving it. For a quadratic polynomial, the discriminant is found using the formula \(D = b^2 - 4ac\).

If the discriminant is positive, there are two distinct real roots. If it is zero, there is one real root, and if negative, there are no real roots in the real number system. The concept of discriminant extends to higher-degree polynomials and is fundamental in determining the nature and count of a polynomial's roots. In our cubic polynomial example, a positive discriminant indicates two critical points, leading to the presence of a local maximum and minimum.

Second Derivative Test

The second derivative test is a powerful tool used to determine if a critical point is a local maximum, local minimum, or neither. Once you've found the critical points of a function, you use the second derivative of the function to assess the concavity at those points.

A positive second derivative at a critical point implies the function is concave up, which usually indicates a local minimum. Conversely, a negative second derivative suggests the graph is concave down, signaling a local maximum. If the second derivative is zero, the test is inconclusive. For the cubic polynomial question, given that the second derivative is linear, one root of the first derivative will correspond to a maximum and the other to a minimum if the discriminant is positive.