Let S: z = 4x^2 + y^2 be the elliptic paraboloid, and let (a, b, c) on S be a po

Question

Let S: z = 4x^2 + y^2 be the elliptic paraboloid, and let (a, b, c) on S be a point closest to (0, 0, 4). Then, we believe that nabla f(a, b, c) =, lambda(a, b, c-4). Consider the case a = 0 and b notequalto 0, and compute the possibilities of (a, b, c). Consider the case a notequalto 0 and b notequalto 0, and compute the possibilities of (a, b, c). Consider the case a = 0 and b = 0, and compute the possibilities of (a, b, c). Considering all the possibilities in the above anlaysis, conclude which one are correct answers.

Explanation / Answer

We are using lagrange multiplier to find the a point (a,b,c) on S which is closet to the point (0,0,4). In order to find the same using this methof we need to take the condtions where a and b equals to 0 simulatneously and any of the variable either a or b equals to zero. Then only we can find the required solutions.

Therfore we need to consider all the posibilities given in (a), (b) and (c). Hence, (d) is the correct option.

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