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Q16E

Expert-verifiedFound in: Page 209

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 \(f(x)\).

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

**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.**

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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