Problem 54 - Calculus A New Horizon

Question

Let \(h\) and \(g\) have relative maxima at \(x_{0} .\) Prove or disprove: (a) \(h+g\) has a relative maximum at \(x_{0}\). (b) \(h-g\) has a relative maximum at \(x_{0}\).

Step-by-Step Solution

Answer

(a) True: \(h+g\) has a relative maximum at \(x_0\). (b) False: \(h-g\) does not necessarily have a relative maximum at \(x_0\).

Step 1: Understanding Relative Maximum

For a function to have a relative maximum at a point \(x_0\), the derivative of the function must be zero at that point, and it must change from positive to negative as it moves past \(x_0\). If we denote the derivative of a function \(f(x)\) by \(f'(x)\), then \(f'(x_0) = 0\) is necessary.

Step 2: Condition for Relative Maxima for h and g

Given that both \(h\) and \(g\) have relative maxima at \(x_0\), we have the conditions \(h'(x_0) = 0\) and \(g'(x_0) = 0\). Additionally, \(h'(x)\) changes from positive to negative through \(x_0\), and the same behavior is expected from \(g'(x)\).

Step 3: Proving (a): Sum of Functions h+g

The function \(h + g\) is defined as \((h+g)(x) = h(x) + g(x)\). The derivative is \((h+g)'(x) = h'(x) + g'(x)\). Since \(h'(x_0) = 0\) and \(g'(x_0) = 0\), it implies \((h+g)'(x_0) = 0 + 0 = 0\). Moreover, \((h+g)'(x)\) will change sign from positive to negative if both \(h'(x)\) and \(g'(x)\) change from positive to negative. Hence, \(h+g\) has a relative maximum at \(x_0\).

Step 4: Proving (b): Difference of Functions h-g

The function \(h - g\) is defined as \((h-g)(x) = h(x) - g(x)\). The derivative is \((h-g)'(x) = h'(x) - g'(x)\). Since \(h'(x_0) = 0\) and \(g'(x_0) = 0\), it implies \((h-g)'(x_0) = 0 - 0 = 0\). However, for \((h-g)(x)\) to have a relative maximum at \(x_0\), \((h-g)'(x)\) must change sign from positive to negative around \(x_0\). Without more information on how \(h'(x)\) and \(g'(x)\) behave together around \(x_0\), this condition cannot be guaranteed. Thus, we cannot conclude \(h-g\) has a relative maximum at \(x_0\).

Key Concepts

Relative Maxima

In calculus, a relative maximum occurs at a particular point of a function, where this point is higher than the points directly surrounding it. Intuitively, it's like being on the top of a hill. More formally, if we have a function \(f(x)\) and it achieves a relative maximum at \(x_0\), then \(f'(x_0) = 0\) is necessary, indicating a horizontal tangent at that point. The derivative, which is the slope, should change from positive on the left-side of \(x_0\) to negative on the right-side. This sign change indicates that the function is increasing up to \(x_0\) and then decreasing past \(x_0\). Such a shift ensures a peak at that point, affirming it's a relative maximum.

Function Derivatives

Derivatives are powerful tools in calculus. They describe how a function changes at every point. Think of the derivative as the rate of change or the slope of the function's graph at any given point. For instance, given a function \(f(x)\), the derivative \(f'(x)\) tells us how \(f(x)\) is changing as \(x\) changes. At points of relative maxima or minima, this rate of change is zero as the slope is horizontal. This means if \(h\) and \(g\) have relative maxima at \(x_0\), then both \(h'(x_0)\) and \(g'(x_0)\) must be zero. Understanding derivatives helps in predicting the behavior of functions around points of interest.

Sum and Difference of Functions

When dealing with the combination of functions, like their sum or difference, derivatives still follow some basic rules. For the sum of functions, such as \( (h+g)(x) = h(x) + g(x) \), the derivative is simply the sum of their individual derivatives, meaning \((h+g)'(x) = h'(x) + g'(x)\). This property aligns neatly with the rule for finding maxima; if both functions have derivatives that change from positive to negative at the same point \(x_0\), the sum also shares this characteristic. On the other hand, the difference of two functions, given by \((h-g)(x) = h(x) - g(x)\), has a derivative of \((h-g)'(x) = h'(x) - g'(x)\). For a relative maximum, the derivative should change from positive to negative, but for differences, there might be complex interactions between \(h'(x)\) and \(g'(x)\) as their behaviors interact. Without further specific conditions, we can't assume a relative maximum at \(x_0\) just based on \(h-g\) because the behavior could vary. Thus, understanding these basics helps us deduce whether a point is a relative maxima or not in complex combinations of functions.