3. Let f be the function defined by
f (x) =( ex/2)/x.
(a) Is there a y–intercept? (Explain)
(b) Determine the horizontal and vertical asymptotes (if any).
(c) Use the sign pattern for f'(x) to determine
(i) the interval(s) over which f rises and where it falls;
(ii) the local extrema.
(d) Use the sign pattern for f
00 (x) to determine.
(i) where the graph of f is concave up and where it is concave down;
(ii) the inflection point(s) (if any). [20]
4. A rectangular sheet of metal of perimeter 36cm and dimensions x and y is to be rolled into a
cylinder. (Volume of cylinder is V = πr2h.) What values of x and y give the largest volume?
(The thickness of the metal can be neglected.)
"\\displaystyle\n(1)\\\\\nf(x) = \\frac{e^{\\frac{x}{2}}}{x}\\\\\n\n(a)\\\\\n\nf(0) = \\infty, \\\\\n\n\\textsf{No}\\,\\, y-\\textsf{intercept}\\\\\n(b)\\\\\n\\textsf{vertical assumptote does not exist since}\\\\\ne^{x\/2} = 0, x \\to -\\infty\\\\\n\nf(\\infty) = 0\\,\\,\\textsf{Horizontal assumptote is}\\,\\, y = 0.\\\\\n\n\n(c)\\\\\nf'(x) = \\frac{1}{2}\\frac{e^{\\frac{x}{2}}}{x} - \\frac{e^{\\frac{x}{2}}}{x^2}\\\\\n\nf(x)\\,\\,\\textsf{rises if}\\,\\, f'(x) > 0\\\\\n\\implies \\frac{xe^{\\frac{x}{2}} - 2e^{\\frac{x}{2}}}{2x^2} > 0\\\\\n\\implies e^{\\frac{x}{2}}(x - 2) > 0 \\\\\n\\implies x > 2\\\\\n\nf(x)\\,\\,\\textsf{falls}\\,\\, f'(x) < 0\\\\\n\\implies \\frac{xe^{\\frac{x}{2}} - 2e^{\\frac{x}{2}}}{2x^2} < 0\\\\\n\\implies e^{\\frac{x}{2}}(x - 2) < 0 \\\\\n\\implies x < 2\\\\\n\n\\textsf{Local extrema}\\,\\, \nf(2) = \\frac{e}{2}\\\\\n\n(d)\\\\\nf(x)\\,\\,\\, \\textsf{is concave up, iff}\\,\\,\\,f''(x) > 0\\\\\n\n\\implies f(x) = \\frac{2x^2e^{\\frac{x}{2}} + x^3e^{\\frac{x}{2}} - 2x^2e^{\\frac{x}{2}} - 4x(x - 2)e^{\\frac{x}{2}}}{4x^4} > 0\\\\\n\\implies x^2 - 4x + 8 > 0\\\\\n\\textsf{The roots of the quadratic}\\\\\n\\textsf{equation are complex}\\\\\n\\textsf{Hence, the function is not concave up}\\\\\n\n\nf(x)\\,\\,\\, \\textsf{is concave down, iff}\\,\\,\\,f''(x) < 0\\\\\n\n\\implies f(x) = \\frac{2x^2e^{\\frac{x}{2}} + x^3e^{\\frac{x}{2}} - 2x^2e^{\\frac{x}{2}} - 4x(x - 2)e^{\\frac{x}{2}}}{4x^4} < 0\\\\\n\\implies x^2 - 4x + 8 < 0\\\\\n\\textsf{The roots of the quadratic}\\\\\n\\textsf{equation are complex}\\\\\n\\textsf{Hence, the function is not concave down}\\\\\n\\textsf{There is inflexion point iff}\\,\\,\\,f''(x) = 0\\\\\n\\textsf{But, since the roots of the}\\\\\n\\textsf{quadratic equation are complex,}\\\\\n\n\\textsf{There are no inflexion points}\\\\\n\n\n(2)\\\\\\textsf{Area of the rectangle} = xy\\\\\n\n\\textsf{Volume of cylinder} = \\pi r^2 h\\\\\n\n\\textsf{A larger surface area gives a}\\\\\n \\textsf{larger volume.}\\\\\n\n2(x + y) = 36, \\, x + y = 18\\\\\n\nA = x(18 - x)\\\\\nA = x(18 - x)\\\\\nA'(x)= 18 - 2x\\\\\n\nA \\,\\, \\textsf{is maximum or minimum at}\\,\\, A'(x) = 0\\\\\n\n\\implies 18 - 2x = 0, x = 9\\\\\n\ny = 18 - x = 18 - 9 = 9\\\\\n\nx = 9cm \\,\\, \\textsf{and}\\,\\, y = 9cm\\,\\, \\textsf{gives the largest volume.}"
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