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Chapter 4: Problem 81

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 \(x_{0}\).

Short Answer

Expert verified
(a) Proven; (b) Disproven.

Step by step solution

01

Understanding Relative Maxima

A function has a relative maximum at a point if the derivative at that point is zero (critical point) and the second derivative is negative. For function, say \(f(x)\), \(f'(x_0) = 0\) and \(f''(x_0) < 0\).
02

Given Conditions

Both \(h(x)\) and \(g(x)\) have a relative maximum at \(x_0\), which implies \(h'(x_0) = 0\), \(h''(x_0) < 0\) and \(g'(x_0)=0\), \(g''(x_0)<0\).
03

Proving (a): Evaluate \\(h+g\\) at \\(x_0\\)

Consider the sum \(f(x) = h(x) + g(x)\). The derivative \(f'(x) = h'(x) + g'(x)\). Evaluating this at \(x_0\), \(f'(x_0) = h'(x_0) + g'(x_0) = 0 + 0 = 0\). The second derivative is \(f''(x) = h''(x) + g''(x)\). Evaluating at \(x_0\), \(f''(x_0) = h''(x_0) + g''(x_0) < 0\) since both are negative. Hence, \(h+g\) has a relative maximum at \(x_0\).
04

Proving (b): Evaluate \\(h-g\\) at \\(x_0\\)

Consider the difference \(f(x) = h(x) - g(x)\). The derivative \(f'(x) = h'(x) - g'(x)\). Evaluating at \(x_0\), \(f'(x_0) = h'(x_0) - g'(x_0) = 0 - 0 = 0\). The second derivative is \(f''(x) = h''(x) - g''(x)\). Evaluating at \(x_0\), \(f''(x_0) = h''(x_0) - g''(x_0) = ext{negative} - ext{negative} = ext{unknown sign}\). This means we cannot conclude if \(h-g\) has a relative maximum at \(x_0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives
Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes.
  • A derivative expresses the rate of change or slope of a function at any given point.
  • If the derivative of a function, expressed as \(f'(x)\), equals zero at a specific point \(x_0\), this indicates that there might be a local extremum (a maximum or a minimum) at that point.
For example, consider functions \(h(x)\) and \(g(x)\) both having relative maxima at \(x_0\). This means their derivatives at \(x_0\), \(h'(x_0)\) and \(g'(x_0)\), are zero. Variations in derivatives are key to identifying not just relative maxima, but also flat areas or inflection points in various functions.
Knowing how to work with derivatives gives a powerful tool for understanding the behaviour of functions in mathematical analysis.
Critical Points
Critical points occur where the derivative of a function is zero or undefined. These points are key in determining where relative maxima or minima might occur.
  • At critical points, the function's rate of change halts, potentially indicating a peak (maximum), a trough (minimum), or an inflection point.
  • For example, if \(h(x)\) and \(g(x)\) reach a relative maximum at \(x_0\), both have their derivatives equal to zero there: \(h'(x_0) = 0\) and \(g'(x_0) = 0\).
It's crucial because, for functions like \(h + g\) or \(h - g\), understanding where the original functions have critical points helps in testing combined functions for extrema. Evaluating these points assists in probing deeper into the functions' behaviour and how they interact at specific values.
Concavity
Concavity refers to the direction in which a curve bends. It is determined using the second derivative of a function:
  • If the second derivative \(f''(x)\) is less than zero at a point, the function is concave down (like a hill) at that point, indicating a possible maximum.
  • If \(f''(x)\) is greater than zero, the function is concave up (like a valley), pointing to a possible minimum.
To decide if functions \(h(x)\) and \(g(x)\) both having maxima means their sum or difference would also have a maximum, examining concavity is critical.
  • For \(h + g\) at \(x_0\), if both second derivatives \(h''(x_0)\) and \(g''(x_0)\) are negative, the combined function is concave down.
  • However, for \(h - g\), each function's concavity may "cancel out," resulting in an uncertain sign for \(f''(x_0)\). This makes it impossible to declare it a maximum.
Understanding these concepts helps in analyzing how functions behave not only individually but also when combined, providing insight into their potential maxima or minima.

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Most popular questions from this chapter

A rectangular poster has one side of length \(x\) inches and total area 700 square inches. The poster must have margins so that the region available for printing has area $$A(x)=728-7 x-\frac{2800}{x} \text { square inches }$$ What are the dimensions of the poster with the greatest printing area?

Determine whether the statement is true or false. Explain your answer. Rolle's Theorem says that if \(f\) is a continuous function on \([a, b]\) and \(f(a)=f(b),\) then there is a point between \(a\) and \(b\) at which the curve \(y=f(x)\) has a horizontal tangent line.

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=\frac{2}{e^{x}+e^{-x}}$$

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=10 \ln x-x$$

Let \(f(x)=a x^{2}+b x+c,\) where \(a>0 .\) Prove that \(f(x) \geq 0\) for all \(x\) if and only if \(b^{2}-4 a c \leq 0 .\) [Hint: Find the minimum of \(f(x) .]\)
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