Question

Find the critical points of $$ f(x)=\frac{x^{3}}{3}+\frac{b x^{2}}{2}+c x+d $$ and identify them as local maxima, local minima, or neither.

Short answer

Question: Determine the nature of the critical points of the function \(f(x) = \frac{x^3}{3} + \frac{b x^2}{2} + cx + d\) using the second derivative test. Answer: The nature of the critical points of the given function cannot be determined explicitly without knowing the values of b and c. However, we can analyze the expression for the second derivative, \(f''(x) = 2x + b\), to determine the nature of the critical points as follows: - If \(2x_0 + b > 0\), there is a local minimum at \(x_0\) - If \(2x_0 + b < 0\), there is a local maximum at \(x_0\) - If \(2x_0 + b = 0\), an additional method will have to be used to determine the nature of the critical point.

Step-by-step solution

Differentiate the function

To find the critical points, first we need to find the derivative of the given function. The given function is: $$ f(x) = \frac{x^3}{3} + \frac{b x^2}{2} + c x + d $$ Now, let's find its first derivative: $$ f'(x) = \frac{d}{dx}\left(\frac{x^3}{3}\right) + \frac{d}{dx}\left(\frac{b x^2}{2}\right) + \frac{d}{dx}(c x) + \frac{d}{dx}(d) $$ $$ f'(x) = x^2 + b x + c $$

Find critical points

A critical point of the function occurs where the derivative is either zero or undefined. As the derivative is a polynomial, it is always defined. Therefore, let's find where it is equal to zero by solving the equation \(f'(x) = 0\): $$ x^2 + b x + c = 0 $$ We can't solve this equation explicitly unless knowing the values of b and c, but we have a critical point where the equation is true.

Compute the second derivative

To determine the nature of the critical points (maxima, minima, or neither), let's calculate the second derivative of the function: $$ f''(x) = \frac{d^2}{dx^2} \left( x^2 + b x + c \right) $$ $$ f''(x) = 2x + b $$

Analyzing critical points using the second derivative test

Now we will use the second derivative test to determine the nature of the critical points. For any critical point \(x_0\), if: - \(f''(x_0) > 0\), then there is a local minimum at \(x_0\) - \(f''(x_0) < 0\), then there is a local maximum at \(x_0\) - \(f''(x_0) = 0\), then the test is inconclusive; other methods (e.g., higher-order derivatives) need to be used to determine the nature of the critical point. As we don't have the explicit critical points, it's impossible to precisely identify the nature of each critical point. However, analyzing the expression for \(f''(x)\): - If \(2x_0 + b > 0\), there is a local minimum at \(x_0\) - If \(2x_0 + b < 0\), there is a local maximum at \(x_0\) - If \(2x_0 + b = 0\), an additional method will have to be used to determine the nature of the critical point.

Key concepts

Derivative of a Function

Understanding the derivative of a function is a cornerstone of calculus and plays a crucial role in finding the critical points of a function. The derivative represents the rate at which the function's value changes concerning its input. It answers the question, 'How quickly is the function's output changing at any specific point?'
When we talk about the derivative of a function like \( f(x) = \frac{x^3}{3} + \frac{b x^2}{2} + c x + d \), we're interested in its instantaneous slope at any point x. To find this, we perform differentiation, which is a process that computes the limit of the average rate of change as the interval over which we measure this rate shrinks to zero.
For polynomials, differentiation is done by applying power rules. Each term of the polynomial is differentiated independently and the constant term (like 'd' in the given function) drops out as its derivative is zero. In the given example, after differentiating term by term, the first derivative \( f'(x) = x^2 + bx + c \) is obtained, which is essential for determining critical points.

Second Derivative Test

The second derivative test is a convenient tool used to determine the concavity of a function at its critical points, which helps in classifying these points as local maxima, minima, or neither. To apply this test, we must first find the second derivative of the function, \( f''(x) \).
In our case, applying the differentiation rules again to the first derivative \( f'(x) = x^2 + bx + c \) gives us the second derivative \( f''(x) = 2x + b \). The second derivative provides us with information about the curvature of the graph of the function: if it's concave up or concave down at the critical points.
If the second derivative at a critical point is positive (\( f''(x_0) > 0 \)), the function is concave up, which means the critical point is a local minimum. Conversely, if the second derivative is negative (\( f''(x_0) < 0 \)), the function is concave down and the point is a local maximum. When \( f''(x_0) = 0 \), the test is inconclusive, and further analysis is required to determine the function's behavior at \( x_0 \).

Local Maxima and Minima

Identifying local maxima and minima is about understanding where a function reaches its highest or lowest value in a neighborhood around a point. A local maximum is a point where the function's value is higher than at any nearby points, and a local minimum is where it's lower than all adjacent points.
To find these points, we look for critical points where the function's derivative equals zero or is undefined. In our function \( f(x) \), after finding the first derivative and setting it to zero, we get the quadratic equation \( x^2 + bx + c = 0 \). The solutions to this equation, if any, are the x-values of the critical points. To classify these points, we use the second derivative test.
By plugging these critical points into the second derivative, we can figure out the nature of each point. However, as real analysis often deals with general cases, we may not always have explicit values for 'b' and 'c.' Even without explicit values, knowing the sign of the second derivative at the critical points allows us to understand whether we're dealing with a local maximum, a local minimum, or if further investigation is needed. This process is invaluable in mapping the behavior of complex functions over different intervals.