91Ó°ÊÓ

Writing In Exercises \(3-6,\) explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b) .\) \(f(x)=\sqrt{\left(2-x^{2 / 3}\right)^{3}},\) \([-1,1]\)

Finding Numbers In find two positive numbers that satisfy the given requirements. The product is 147 and the sum of the first number plus three times the second number is a minimum.

Using Rolle's Theorem In Exercises \(7-10\) , find the two \(x\) -intereepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. \(f(x)=x^{2}-x-2\)

Using Newton's Method In Exercises \(7-16,\) use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}+4\)

Using a Tangent Line Approximation In Exercises \(5-10\) , find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Then complete the table. \(f(x)=x^{5}, \quad(2,32)\)

Determining Concavity In Exercises \(5-16,\) determine the open intervals on which the graph of the function is concave upward or concave downward. \(f(x)=x^{4}-3 x^{3}\)

Using Rolle's Theorem In Exercises \(7-10\) , find the two \(x\) -intereepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. \(f(x)=x^{2}+6 x\)

Determining Concavity In Exercises \(5-16,\) determine the open intervals on which the graph of the function is concave upward or concave downward. \(h(x)=x^{5}-5 x+2\)

Using a Tangent Line Approximation In Exercises \(5-10\) , find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Then complete the table. \(f(x)=\sqrt{x}, \quad(2, \sqrt{2})\)

Using Newton's Method In Exercises \(7-16,\) use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=2-x^{3}\)