91Ó°ÊÓ

True or False? In Exercises \(53-56,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Find the area of the largest rectangle that can be inscribed under the curve \(y=e^{-x^{2}}\) in the first and second quadrants.

The profit \(P\) (in thousands of dollars) for a company spending an amount \(s\) (in thousands of dollars) on advertising is \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\) (a) Find the amount of money the company should spend on advertising in order to obtain a maximum profit. (b) The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns.

Find \(a, b, c,\) and \(d\) such that the cubic \(f(x)=a x^{3}+b x^{2}+c x+d\) satisfies the given conditions. Relative maximum: (2,4) Relative minimum: (4,2) Inflection point: (3,3)

A section of highway connecting two hillsides with grades of \(6 \%\) and \(4 \%\) is to be built between two points that are separated by a horizontal distance of 2000 feet (see figure). At the point where the two hillsides come together, there is a 50 -foot difference in elevation. (a) Design a section of highway connecting the hillsides modeled by the function \(f(x)=a x^{3}+b x^{2}+c x+d\) \((-1000 \leq x \leq 1000)\). At the points \(A\) and \(B,\) the slope of the model must match the grade of the hillside. (b) Use a graphing utility to graph the model. (c) Use a graphing utility to graph the derivative of the model. (d) Determine the grade at the steepest part of the transitional section of the highway.

The deflection \(D\) of a beam of length \(L\) is \(D=2 x^{4}-5 L x^{3}+3 L^{2} x^{2},\) where \(x\) is the distance from one end of the beam. Find the value of \(x\) that yields the maximum deflection.

Use symmetry, extrema, and zeros to sketch the graph of \(f .\) How do the functions \(f\) and \(g\) differ? Explain. $$ f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}, \quad g(x)=x\left(x^{2}-3\right) $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of \(f(x)=1 / x\) is concave downward for \(x<0\) and concave upward for \(x>0\), and thus it has a point of inflection at \(x=0\)

Let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\). Prove that if \(f\) and \(g\) are positive, increasing, and concave upward on the interval \((a, b),\) then \(f g\) is also concave upward on \((a, b)\).

Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=f(x-10) \quad g^{\prime}(0) \quad 0 $$