Question 1089274: p(a,b) is a variable point in the first quadrant on the curve y=e^-x. Q and R are the points where the tangent to the curve at p meets the x- and y - axes respectively. find the maximum area of the triangle OQR, where O is the origin.
thank you
Answer by natolino_2017(77)   (Show Source):
You can put this solution on YOUR website! y = exp(-x)
b = exp(-a).
dy/dx = -exp(-x), so dy/dx = -exp(-a) = -b (slope on the tangent line on (a,b))
Let H: height of the triangle
L: Base of the triangle.
we need H in the terms of a and L in the terms of a.
-b = (H-b)/(0-a). So H(a) = ab+b = exp(-a)(a+1).
-b = (b-0)/(a-L). So L(a) = a+1
A(a) = H(a)*(L(a))/2 (Area of the triangle: Base*height/2)
A(a) = exp(-a)(a+1)^2/2 with a>0
A'(a) = 0.5(-exp(-a)(a+1)^2 +2*exp(-a)(1+a)) = 0
solving for a: a = 1.
A''(a) = 0.5(-exp(-a)(1-a^2)-2aexp(-a)) = 0.5(exp(-a)(a^2-2a-1)
A''(1) = -exp(-1) < 0, so the point is a max
A(1) = 0.5exp(-1)(1+1)^2 = 2exp(-1) [unit^2] (0.736 [unit^2] aprox)
@natolino_
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