Find the absolute maximum and minimum values of the function on the given curve.
-$$\text { Function: } f ( x , y ) = x ^ { 2 } + y ^ { 2 } ; \text { curve: } x = 5 t + 1 , y = 5 t - 1,0 \leq t \leq 1 \text {. }$$
A) Absolute maximum: 52 at t = 1; absolute minimum: 2 at t = 0.
B) Absolute maximum: 51 at t = 1; absolute minimum: 6 at t = 0.
C) Absolute maximum: 51 at t = 1; absolute minimum: 2 at t = 0.
D) Absolute maximum: 52 at t = 1; absolute minimum: 6 at t = 0.

Question

Quizplus · Accepted Answer

Substituting $x = 5t + 1$ and $y = 5t - 1$ into $f(x, y) = x^2 + y^2$ gives $f(t) = (5t + 1)^2 + (5t - 1)^2$. At $t = 0$, $f(0) = 1^2 + (-1)^2 = 2$. At $t = 1$, $f(1) = (5*1 + 1)^2 + (5*1 - 1)^2 = 6^2 + 4^2 = 36 + 16 = 52$.