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Essential Calculus: Early Transcendentals
Found in: Page 209
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\). (Use the graphs and transformations of Sections 1.2.)

16. \(f(x) = 2 - \frac{1}{3}x,\;x \ge - 2\)

The absolute maximum of \(f(x)\) is \(\frac{8}{3}\). There is no absolute minimum for \(f(x)\). There is no local maximum and local minimum for \(f(x)\).

See the step by step solution

Step by Step Solution

Step 1: Given data

The given function is \(f(x) = 2 - \frac{1}{3}x,x \ge - 2\).

Step 2: Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Step 3: Sketch the graph

It is given that, \(y = 2 - \frac{1}{3}x,x \ge - 2\).

Express the given equation in the intercept form.

\(\begin{aligned}{c}y &= 2 - \frac{1}{3}x\\x + 3y &= 6\\\frac{x}{6} + \frac{y}{{\left( {\frac{6}{3}} \right)}} &= \frac{6}{6}\\\frac{x}{6} + \frac{y}{2} &= 1\end{aligned}\)

Here, the \(x\)-intercept is 6 and the \(y\)-intercept is 2.

Plot the graph:

Since the condition of the function is \(x \ge - 2\), draw the line graph up to \(x = - 2\).

From Figure, it is observed that the graph \(f(x)\) attains the maximum value at \(x = - 2\).

Substitute \( - 2\) in the original function \(f(x)\) and obtain the value of absolute maximum.

\(\begin{aligned}{c}f( - 2) &= 2 - \frac{1}{3}( - 2)\\f( - 2) &= 2 + \frac{2}{3}\\f( - 2) &= \frac{8}{3}\end{aligned}\)

Thus, the absolute maximum of \(f(x)\) is \(\frac{8}{3}\) . Also it is observed from Figure, that there is no absolute minimum for \(f(x)\) since the straight line approaches to \( - \infty \). The function is decreasing in the entire domain. So, there is no local maximum and local minimum.

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