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Chapter 4: Q10E (page 208)

Sketch the graph of a function that is continuous on (1, 5) and has the given properties.

10. \(f\) has no local maximum or minimum but 2 and 4 are critical numbers.

Short Answer

Expert verified

The graph is sketched.

Step by step solution

01

Given data

The given function is the critical numbers 2 and 4.

02

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.

03

Sketch the graph

Given that the graph of a function that is continuous on the interval (1,5).

The graph as critical point at \(x = 2\) and \(x = 4\).

Consider the function \(f(x) = \sqrt(3){{(x - 2)(x - 4)}}\) for \((1,2) \cup (4,5)\).

Differentiate the function with respect to \(x\) as follows.

\({f^\prime }(x) = \frac{{\sqrt(3){{(x - 4)}}}}{{3{{(x - 2)}^{\frac{2}{3}}}}} + \frac{{\sqrt(3){{(x - 2)}}}}{{3{{(x - 4)}^{\frac{2}{3}}}}}\)

Since the function obtained above is not defined for \(x = 2\) and \(x = 4\), 2 and 4 are in the domain of \(f(x)\), then \(x = 2\) and \(x = 4\) are critical numbers. The graph of the given function is:

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Most popular questions from this chapter

Find the critical numbers of the function.

35.\(f(x) = {x^2}{e^{ - 3x}}\).

Find the absolute maximum and absolute minimum values of on the given interval.

46. \(f(t) = t + \cot \left( {\frac{t}{2}} \right)\), \(\left( {\frac{\pi }{4},\frac{{7\pi }}{4}} \right)\)

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.)

18. \(f(t) = \cos t,\;\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\)

(a) Show that \({e^x} \ge 1 + x\) for \(x \ge 0\).

(b) Deduce that \({e^x} \ge 1 + x + \frac{1}{2}{x^2}\) for \(x \ge 0\).

(c) Use mathematical induction to prove that for \(x \ge 0\) and any positive integer \(n\),

\({e^x} \ge 1 + x + \frac{{{x^2}}}{{2!}} + \ldots \ldots + \frac{{{x^n}}}{{n!}}\)

Determine a cubic function with the given criteria.
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