​ (a) Find the equation of the tangent to the curve x^2/3 + y^2/3 = 2 at (1,1). ​

Question

(a) Find the equation of the tangent to the curve ^{\(x^\frac{2}{3}\)} + ^{\(y^\frac{2}{3}\)} = 2 at (1,1).

(b) Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

Answer 1

(a) Given; \(x^\frac{2}{3} + y^\frac{2}{3}\) = 2 ; differentiating w.r.t to x;

(b) Let the numbers be x, 15 - x.

S (x) = x² + (15- x)² = 2x² - 30x + 225

For turning points;

S'(x) = 4x - 30 = 0 

S" (x) = 4 > 0 Therefore maximum

∴ x = \(\frac{15}{2}\) is the point of local maximum of s.

The numbers are \(\frac{15}{2}\), \(\frac{15}{2}\).