Question

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=x^{3}-\frac{3}{2} x^{2}-6 x $$

Short answer

After performing these steps, we have calculated the first and second derivatives and solved them to find potential critical points and points of inflection. The critical points and points of inflection can further be confirmed by graphing the function, seeing where high points and low points exist within local areas (those will be the extrema), and noting where the curvature of the function changes (those will be the points of inflection).

Step-by-step solution

Calculate the first derivative

The first thing to do is to calculate the derivative of \(f(x)\), i.e, \(f'(x)\). This can be done using the power rule of differentiation.\n\nThe derivative of \(x^n\) with respect to \(x\) is \(n*x^{n-1}\). The derivative of a constant is zero.\n\nTherefore, the derivative of \(f(x) = x^3 - \frac{3}{2}x^2 - 6x\) is: \n\n\(f'(x) = 3*x^{3-1} - \frac{3}{2}*2*x^{2-1} - 6 = 3x^2 - 3x - 6\).

Find critical points

The critical points of function \(f(x)\) are those values of x at which the first derivative of the function equals zero, (\(f'(x) = 0\)), or does not exist.\n\nThe derivative of \(f(x)\) is a quadratic equation which can be solved by setting it to zero, \(3x^2 - 3x - 6=0\). Solving for \(x\), we have the critical points.\n\nNote: This quadratic equation can be solved using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where a, b, and c are the coefficients of the equation \$ax^2 + bx + c = 0.

Calculate the second derivative

To identify points of inflection, we need to calculate the second derivative, \(f''(x)\), and set it to zero to find possible points of inflection.\n\nUsing the power rule of differentiation again, the second derivative of \(f(x)\) is \(f''(x) = 6x - 3\).

Find points of inflection

Points of inflection are values of x at which the second derivative of the function equals zero, (\(f''(x) = 0\)), or does not exist.\n\nSet \(f''(x) = 6x - 3 = 0\). Solving this, we get a possible point of inflection. We need to check if the concavity changes on either side of this point. If it does, then it's a point of inflection.

Graph the function and identify relative extrema and points of inflection

After the critical points and potential points of inflection have been found, use a graphing utility to graph the function. The relative extrema are the highest and lowest points on the graph in a local neighborhood, while inflection points are where the graph changes curvature.

Key concepts

Relative Extrema

Understanding the concept of relative extrema is crucial when examining the behavior of functions in calculus. Relative extrema, also known as local maxima and minima, are points where a function reaches a peak or a valley on a certain interval. To find the relative extrema, we examine the first derivative, which tells us the slope of the tangent line at any given point.

Where the derivative is zero, the slope is flat, and these points are potential candidates for being local maxima or minima. However, further investigation with the second derivative or a sign chart is necessary to confirm whether these are indeed relative maxima or minima. In the given exercise, the derivative calculation is our first step towards identifying these points.

Points of Inflection

A central concept in the study of a function's graph is the point of inflection. A point of inflection is a location on the graph where the concavity of the function changes. This means that the curve goes from being 'smiley' (concave up) to 'frowny' (concave down), or vice versa.

To identify these points, we calculate the second derivative of the function because it provides information about the concavity. If the second derivative changes sign around a point, that point is a potential inflection point. The exercise guides us through the computation of the second derivative, which sets the stage for locating points where the function changes its curvature.

Derivative Calculations

The foundation of many calculus problems involves derivative calculations. The derivative of a function at a particular point gives us the rate of change of the function at that point. It is mathematically defined as the limit of the average rate of change as the interval shrinks to zero.

For polynomial functions, like in our exercise, we often use the power rule for differentiation, which makes these calculations straightforward. Once we have the first derivative, we can find critical points where the derivative is zero or undefined—these are of particular interest for finding relative extrema and potential points of inflection.

Quadratic Equation

The quadratic equation is a standard form of polynomial characterized by its highest degree of x being two (i.e., it is of the form ax² + bx + c). Solving quadratic equations is a fundamental skill in mathematics, and it becomes particularly handy in calculus when finding critical points of a function.

As shown in the step-by-step solution, critical points can be found by setting the first derivative equal to zero, which often results in a quadratic equation. There are various methods to solve for x, but the quadratic formula is a reliable tool, guaranteeing the solution for all quadratic equations.

Second Derivative Test

The second derivative test is a practical method used to determine whether a critical point is a local maximum, a local minimum, or neither. By taking the second derivative of the function and evaluating it at the critical points (where the first derivative is zero), we can discern the concavity at those points.

If the second derivative is positive, the function is concave up, and we have a local minimum. Conversely, if the second derivative is negative, the function is concave down, and it is a local maximum. If the second derivative at the critical point is zero, the test is inconclusive, and further methods must be employed to categorize the critical point.