Suggested languages for you:

Americas

Europe

Q16E

Expert-verified
Found in: Page 209

Essential Calculus: Early Transcendentals

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

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 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.