The curve is defined for .

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The curve x+y=4\sqrt{x}+\sqrt{y}=4 is defined for x≥0, y≥0x\ge 0,\,y\ge 0. (i) Show that dydx=−yx\dfrac{dy}{dx}=-\dfrac{\sqrt{y}}{\sqrt{x}}. (ii) Prove that the curve possesses exactly one horizontal tangent and exactly one vertical tangent, and determine their coordinates. (iii) Write the equations of the horizontal and vertical tangents obtained in part (ii). (iv) For 0<x<160<x<16, discuss the sign (positive or negative) of the slope dydx\dfrac{dy}{dx} and hence comment on the overall orientation of the curve in this interval.

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