Question

Suppose is a differentiable function of one variable. Show that all tangent planes to the surface z = y f(x/y) intersect at a common point.

Answer 1

Given f is a diffentiable Funcholy of one variable. Show that all tongent planes to the surface a=4f(2) insult of q Common point. the equation of the surface is a=lf (2) Hecky let g(x) = how o en su y) = x19.565) Yf6)-0 differentiate with respect to a 940414)= Y F'()() 9(19) < $$1$)) gxilly) = f( ) ax 6)=1) ola now differentiate egin with respect to y (x UV = uv'tvo!) 96ry) hy (4. f(4)) 5.3 (4) + f() - 3 f'B, C + +4) 9) %+4)*71) + +14) 14(219) = f(*)-($$$) now the ea" of the tongent plane at point (2o, Yo, zo) 2-703 fe (Mo, Yo )61-ko)+ify(190, 40) (Y-yo) here fx = 3x q fy_gy + 9 is 22-20= t' tony (470) +(5)** )=($) Jex-xo), now Substitute (1,4,7)=(0,0,0) 의 -20= pre ) (0-10)+ (School ( 9 ) f(!)) (0-40 -1,$'hos) +-y)f("%)+ York *)6.40

-70= -X +(*%*) -40F (201) + Toffeeting +70 Yoflto Yo f (no Yo 3 --> 0= gxry) yfly) hare g(x,y)=2 z = y(1) put (20,9o, zo) we have 206 Zo = Yof (*%o) 8&o are same hence all tangent planes to the given surface are interst 사 a Common point,