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Chapter 4: Problem 41

. Prove that, if \(f\) is continuous on \(I\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x) \geq 0\) on the interior of \(I\), then \(f\) is nondecreasing on I. Similarly, if \(f^{\prime}(x) \leq 0\), then \(f\) is nonincreasing on \(I\).

Short Answer

Expert verified
If \( f' \geq 0 \), then \( f \) is nondecreasing; if \( f' \leq 0 \), then \( f \) is nonincreasing.

Step by step solution

01

Understand Definitions

To prove that a function is nondecreasing on an interval, you must show that for any two points \( x_1 \) and \( x_2 \) in the interval such that \( x_1 < x_2 \), it follows that \( f(x_1) \leq f(x_2) \). Similarly, a nonincreasing function requires \( f(x_1) \geq f(x_2) \) for \( x_1 < x_2 \).
02

Mean Value Theorem Application

The Mean Value Theorem states that if \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a point \( c \in (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). We can use this theorem if \( f' \geq 0 \) and conclude that \( f(b) - f(a) \geq 0 \), implying \( f(a) \leq f(b) \).
03

Proving Nondecreasing Behavior

Assume \( x_1 < x_2 \) and apply the Mean Value Theorem on the interval \([x_1, x_2]\). Since \( f'(x) \geq 0 \) for every \( x \) in \((x_1, x_2)\), there exists a \( c \in (x_1, x_2) \) such that \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \geq 0 \). Therefore, \( f(x_1) \leq f(x_2) \), hence \( f \) is nondecreasing on \( I \).
04

Analogous Argument for Nonincreasing Behavior

If \( f'(x) \leq 0 \) for all \( x \) in the interior of \( I \), then by applying the Mean Value Theorem on the interval \([x_1, x_2] \) with \( x_1 < x_2 \), we have \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \leq 0 \). This implies \( f(x_1) \geq f(x_2) \), and thus \( f \) is nonincreasing on \( I \).

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Key Concepts

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

Mean Value Theorem
The Mean Value Theorem (MVT) is a fundamental concept in calculus that connects the average rate of change of a function over an interval to an instantaneous rate of change within that interval. Imagine driving a car from point A to point B. If you know your average speed was 50 km/h, the Mean Value Theorem assures you that at some moment during your trip, your speedometer showed exactly 50 km/h. This is captured by the theorem's statement:
  • If a function \( f \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \( c \) in \((a, b)\) such that
\[f'(c) = \frac{f(b) - f(a)}{b - a}\]This result is incredibly handy for making inferences about the behavior of a function based on its derivative. If the derivative \(f'(x)\) is always nonnegative or nonpositive in an interval, it helps us draw conclusions about the function's increasing or decreasing nature over that interval.
Continuity in Functions
Continuity is like a magical adhesive that keeps a function glued smoothly together in a given interval. What does that mean? Simply put: a function \( f \) is continuous at a point \( x = c \) if there is no abrupt jump at \( c \). In more technical terms:
  • The limit as \( x \) approaches \( c \) of \( f(x) \) equals \( f(c) \) itself.
  • This must hold true for every point within the interval \([a, b]\).
Why is continuity important? Continuous functions promise reliability鈥攖hey don't suddenly change value without warning. This smooth behavior is crucial when applying the Mean Value Theorem, as it requires the function to be seamless over the interval in question. Imagine knitting a sweater; any knots disrupt the smooth texture much like discontinuities disrupt a function's behavior.
Nondecreasing and Nonincreasing Functions
Understanding nondecreasing and nonincreasing functions is all about how the function behaves as you move along the x-axis:
  • **Nondecreasing Functions:** For any two points \( x_1 \) and \( x_2 \) within the interval, if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). This means that the function doesn't decrease as you go from left to right鈥攊t either stays level or increases.
  • **Nonincreasing Functions:** Similarly, if \( x_1 < x_2 \), we have \( f(x_1) \geq f(x_2) \). The function either stays the same or decreases, but it never increases.
These concepts are linked to the function's derivative. If \( f'(x) \geq 0 \) throughout an interval, the function is nondecreasing. If \( f'(x) \leq 0 \), the function is nonincreasing. This understanding helps in predicting and explaining phenomena where function values represent quantities like cost, speed, or time spent.

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Most popular questions from this chapter

36\. Show that the differential equation $$ \frac{d y}{d t}=a y+b, y(0)=y_{0} $$ has solution $$ y=\left(y_{0}+\frac{b}{a}\right) e^{a t}-\frac{b}{a} $$ Assume that \(a \neq 0\).

Brass is produced in long rolls of a thin sheet. To monitor the quality, inspectors select at random a piece of the sheet, measure its area, and count the number of surface imperfections on that piece. The area varies from piece to piece. The following table gives data on the area (in square feet) of the selected piece and the number of surface imperfections found on that piece. $$ \begin{array}{ccc} \hline \text { Piece } & \begin{array}{c} \text { Area in } \\ \text { Square Feet } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Surface Imperfections } \end{array} \\ \hline 1 & 1.0 & 3 \\ 2 & 4.0 & 12 \\ 3 & 3.6 & 9 \\ 4 & 1.5 & 5 \\ 5 & 3.0 & 8 \\ \hline \end{array} $$ (a) Make a scatter plot with area on the horizontal axis and number of surface imperfections on the vertical axis. (b) Does it look like a line through the origin would be a good model for these data? Explain. (c) Find the equation of the least-squares line through the origin. (d) Use the result of part (c) to predict how many surface imperfections there would be on a sheet with area \(2.0\) square feet

A block slides down an inclined plane with a constant acceleration of 8 feet per second per second. If the inclined plane is 75 feet long and the block reaches the bottom in \(3.75\) seconds, what was the initial velocity of the block?

Prove the formula $$ \int\left[f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right] d x=f(x) g(x)+C $$ Hint: See the box in the margin next to Theorem A.

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as $y^{\prime} / y Prove that if the relative rate of change is a negative constant then the function must represent exponential decay.
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