A rectangle in the figure is formed with adjacent sides on the coordinate axes a
ID: 2690016 • Letter: A
Question
A rectangle in the figure is formed with adjacent sides on the coordinate axes and one corner on the graph of y= 14/((x^2)+1) Find the maximum possible area of this rectangle.Explanation / Answer
Find the coordinate of the corner: (x, f(x)) --> this makes the "width" x and the "height" y (i.e. f(x)), so the area is: A = x * y = x * f(x) = 14 * x / (x² + 1) --> just maximize x / (x² + 1) to find x, then find A later g(x) = x / (x² + 1) --> take derivative--use quotient rule g'(x) = { (x² + 1) - 2x * x) / (x² + 1)² --> simplify numerator x² + 1 - 2x² = 1 - x² = (1 + x)(1 - x) --> g'(x) = (1 + x)(1 - x) / (x² + 1)² --> denominator is NEVER zero, so just set numerator to 0 (1 + x)(1 - x) = 0 --> two roots x = ±1 This should be no surprise, because the function was symmetrical, so there is an extrema on either side of the graph (i.e. positive and negative x values). The negative is likely the "minimum" (because it will be negative) and the positive is clearly the maximum. So the maximum area should be @ x = 1 --> f(1) = 14 / (1² + 1) = 14/2 = 7 --> A = 1 * 7 = 7