Problem 29 - Applied Calculus

Question

Find the value(s) of \(x\) that give critical points of \(y=\) \(a x^{2}+b x+c\), where \(a, b, c\) are constants. Under what conditions on \(a, b, c\) is the critical value a maximum? A minimum?

Step-by-Step Solution

Answer

The critical point is \(x = -\frac{b}{2a}\). It's a minimum if \(a > 0\) and a maximum if \(a < 0\).

Step 1: Find the derivative

To find the critical points of the function, we first need to find its derivative. The function given is \( y = ax^2 + bx + c \). The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = 2ax + b \).

Step 2: Set the derivative to zero

Critical points occur where the derivative is zero or undefined. In this case, we set the derivative equal to zero: \( 2ax + b = 0 \).

Step 3: Solve for \(x\)

Solve the equation \( 2ax + b = 0 \) for \( x \). This gives \( x = -\frac{b}{2a} \). This is the critical point.

Step 4: Determine if the critical point is a maximum or minimum

To determine whether the critical point is a maximum or minimum, examine the second derivative of the function. Compute the second derivative: \( \frac{d^2y}{dx^2} = 2a \).

Step 5: Use the second derivative test

The second derivative test helps to identify the nature of the critical point: \( \frac{d^2y}{dx^2} > 0 \) implies a local minimum, whereas \( \frac{d^2y}{dx^2} < 0 \) implies a local maximum. So, for \( a > 0 \), the point is a minimum, and for \( a < 0 \), it is a maximum.

Step 6: Examine additional conditions

For this quadratic function, the critical point reflects a parabola turning point. If \( a = 0 \), then the function is linear, and there are no maxima or minima defined by a quadratic's vertex. Hence, \( a eq 0 \) for the analysis to be valid.

Key Concepts

Quadratic Functions

Quadratic functions are polynomials of degree two, commonly expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions are known for their U-shaped or inverted U-shaped graphs, called parabolas.

The coefficient \( a \) determines the direction of the parabola:

• If \( a > 0 \), the parabola opens upwards, creating a U-shape.
• If \( a < 0 \), the parabola opens downwards, forming an inverted U-shape.

The terms \( bx \) and \( c \) affect the position and the orientation of the parabola, shifting it along the x-axis or y-axis, respectively. Understanding quadratic functions is key for exploring their properties, such as critical points and vertexes, which are essential in optimization scenarios.

Derivatives

Derivatives provide a way to determine the rate of change of a function’s output with respect to changes in its input. For a quadratic function, the derivative allows us to identify points where the slope of the tangent to the curve is zero or undefined.

Given the function \( y = ax^2 + bx + c \), the derivative with respect to \( x \) is calculated as:\[ \frac{dy}{dx} = 2ax + b \]Setting \( \frac{dy}{dx} = 0 \) helps us find the critical points, where the curve has horizontal tangents. Such points are significant in identifying the function’s local maxima or minima.By solving the equation \( 2ax + b = 0 \), we find the critical point at \( x = -\frac{b}{2a} \). This particular \( x \)-value is essential for further analysis using the second derivative test.

Second Derivative Test

The second derivative test is a method used to classify critical points, discovered through setting the first derivative to zero. For a deeper analysis, the second derivative of the function is used to assess the concavity of the graph at these points.

For the quadratic function \( y = ax^2 + bx + c \), the second derivative is:\[ \frac{d^2y}{dx^2} = 2a \]

• If \( \frac{d^2y}{dx^2} > 0 \), the function has a local minimum – the parabola is concave upwards.
• If \( \frac{d^2y}{dx^2} < 0 \), the function has a local maximum – the parabola is concave downwards.

The sign of \( 2a \) indicates whether the critical point \( x = -\frac{b}{2a} \) corresponds to a local maximum or minimum.

Maximum and Minimum Values

In the context of quadratic functions, finding maximum and minimum values at critical points is crucial for optimization. These values give insights into the highest or lowest points that a quadratic function might reach on its graph.

• A **local maximum** occurs when the parabola opens downwards, meaning \( a < 0 \) and the critical point is at the peak of the parabola.
• A **local minimum** occurs when the parabola opens upwards, meaning \( a > 0 \), and the critical point is at the lowest point.

The critical value found at \( x = -\frac{b}{2a} \) is paramount in determining whether it's a maximum or minimum. When analyzing quadratic functions, always ensure \( a eq 0 \) to avoid the function turning linear, which would not have a defined maximum or minimum.