Question

True or False If \(f^{\prime}(c)=0\) and \(f(c)\) is not a local maximum, then \(f(c)\) is a local minimum. Justify your answer.

Short answer

False. A point where the derivative of a function is zero and is not a local maximum can be a local minimum, or it can also be a saddle point as seen in the case of the function \(f(x) = x^3\) at \(x=0\). Thus, the statement is not necessarily true.

Step-by-step solution

Understand the question

The problem is requesting to determine the veracity and justification of a statement. The statement is saying that if the derivative of a function at a certain point is zero and this point is not a local maximum, then this point must necessarily be a local minimum.

Analyze and validate the premise

A derivative of zero at a point implies that the tangent to the function at that point is horizontal. The fact that this point is not a local maximum is also explicitly mentioned. However, the absence of a local maximum does not automatically infer the presence of a local minimum. Consider the function \(f(x) = x^3\). The derivative is \(f'(x) = 3x^2\), which equals zero when \(x=0\). However, \(f(0)\) is neither a local maximum nor a local minimum, but a saddle point.

Provide the answer

Putting together the findings from the previous steps, it can be concluded that the statement is false. Even if the derivative of a function at a point is zero and if this point is not a local maximum, it is not necessary that this point has to be a local minimum. It can also be a saddle point.

Key concepts

Derivative of a Function

When exploring the concept of the derivative of a function, we're delving into one of the fundamental tools in calculus. The derivative measures how a function changes as its input changes. In simpler terms, it expresses the rate of change or slope of the function at any given point. To calculate the derivative of a function, we may use several techniques, such as the power rule, product rule, quotient rule, or chain rule, depending on the form of the function.

Mathematically, if you have a function denoted by \(f(x)\), the derivative of that function at a point \(x = a\) is represented as \(f'(a)\), which is the slope of the tangent line to the function at that point. The tangent line is a straight line that just

Saddle Point

A saddle point is a critical concept in understanding function behavior, particularly when assessing points where the derivative equals zero. A saddle point, in the realm of calculus, refers to a point on the graph of a function where the derivative is zero but is neither a local maximum nor a local minimum. To identify a saddle point, you examine the changes in the slope (derivative) before and after the point. If the derivative changes sign, indicating a change in the direction of the slope, a local extremum may be present. In contrast, if the slope remains consistent on either side of the point, a saddle point may exist.

A classic example is the function \(f(x) = x^3\), where \(f'(x) = 3x^2\). When \(x = 0\), the derivative is zero. However, despite the slope (derivative) being zero at that point, the nature of the graph around the point resembles a saddle – it does not form a peak or a valley, but instead has a slope that changes direction. In other words, the function transitions from increasing to decreasing or vice versa, but without defining a clear local maximum or minimum.

Local Maximum and Minimum

The terms local maximum and minimum denote points where a function reaches its highest or lowest value, respectively, in a nearby region. To be precise, a local maximum is a point where the function value is greater than (or equal to) the function value at nearby points. Similarly, a local minimum is where the function value is less than (or equal to) those at neighboring points.

One can identify possible local maximums or minimums by finding points where the derivative of the function is zero (\(f'(x) = 0\)) or undefined. These points are called critical points. Once critical points are determined, you can use tests, like the first derivative test or the second derivative test, to confirm whether these points are actually local maxima or minima. With the first derivative test, you examine the sign of the derivative just before and just after the critical point. If the derivative changes from positive to negative, the function has a local maximum. If it goes from negative to positive, it's a local minimum.

Nevertheless, it's essential to note that not every point where the derivative is zero will be a local extremum. As we've discussed in the section on saddle points, a zero derivative could also mean the presence of a saddle point.