Question

True or False If \(f(c)\) is a local maximum of a continuous function \(f\) on an open interval \((a, b),\) then \(f^{\prime}(c)=0 .\) Justify your answer.

Short answer

The statement is False. There exist functions with local maxima at which the derivative either does not exist or is not zero.

Step-by-step solution

Analyze the Relationship

The relationship between a local maximum of a function and its derivative at that point is given by Fermat's Theorem, which states that if \(f(c)\) has a local maximum or minimum and if \(f^{\prime}(c)\) exists, then \(f^{\prime}(c)=0\). But the derivative does not always exist at a local maximum of a function, which is important to consider.

Counterexamples

It's also helpful to think of a function for which the derivative does not exist at a local maximum. An example of this is the absolute value function, \(f(x)= |x|\). This function has a local maximum at \(x=0\), but the derivative, \(f^{\prime}\), does not exist at that point. This is because the function is not differentiable at \(x=0\). There's another example in \(f(x) = x^3\) at \(x=0\). Despite the point \(x=0\) being a local maximum, \(f'(0) = 0\) and not necessarily equal to 0.

Conclusion

Considering this information, the statement, 'If \( f(c) \) is a local maximum of a continuous function \( f \) on an open interval \( (a, b) \), then \( f^{\prime}(c)=0 \),' is not always true. Therefore, the answer is false.

Key concepts

Fermat's Theorem

When studying local maximum calculus, one encounters Fermat's Theorem, a fundamental principle that lays the groundwork for understanding the critical points of functions. This theorem states that if a function has a local maximum or minimum at a point, and if the function is differentiable at that point, then the derivative at that point is zero. The intuitive idea here is that the slope of the tangent line to the function at the peak (maximum) or trough (minimum) of a curve must be flat, which means having a slope of zero.

However, it is crucial to mention that Fermat's Theorem applies only when the function is differentiable at that point. That means a function might still have a local maximum even if the derivative does not exist at that point, complicating the relationship between local extrema and derivatives.

Derivative of a Function

Understanding the derivative of a function is vital in calculus, as it measures how a function changes as its input changes. In essence, the derivative represents the slope of the tangent line to the curve of the function at a certain point. It is the limit of the average rate of change of the function over an interval as that interval becomes infinitesimally small.

Mathematically, if you have a function given by \(f(x)\), its derivative \(f'(x)\) at a certain point \(x = a\) is given by the limit \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]. If this limit exists, we say the function is differentiable at that point and the derivative there gives us essential information about the function's behavior.

Continuous Function

The concept of a continuous function is integral to calculus and the analysis of real-valued functions. A function \(f(x)\) is considered continuous at a point \(x = a\) if three conditions are met: firstly, \(f(a)\) must be defined; secondly, \(\lim_{x \to a} f(x)\) must exist; and thirdly, \(\lim_{x \to a} f(x) = f(a)\). This ensures that the graph of \(f\) at \(a\) has no breaks, jumps, or holes, and can be drawn without lifting the pen from the paper.

A continuous function over an open interval \( (a, b) \) means that the function is continuous at every point within that interval. This continuity is a prerequisite for applying Fermat's Theorem and hence, related to finding local maxima.

Differentiability

Differentiability is a property of a function that tells us if the function's derivative exists at a given point. When a function \(f\) is differentiable at \(x = c\), we can determine its instantaneous rate of change at that point using its derivative \(f'(c)\). Differentiability implies continuity; if a function is differentiable at a certain point, it must also be continuous there. However, the converse is not always true; a function can be continuous at a point but not differentiable there.

For instance, a function with a sharp corner or cusp, like the absolute value function at \(x = 0\), is continuous but not differentiable at that point. Identifying whether a function is differentiable at a point is an important step in analyzing its local extrema and applying Fermat's Theorem.