Question
4) Three friends are working on a problem concerning a differentiable function f(x,y,z) at a point...
4) Three friends are working on a problem concerning a differentiable function f(x,y,z) at a point P. While these 3 people are discussing their calculations: 1st person says: "I found the equation of the tangent plane to the surface f(x, y, z)=c at the point P to be 3x + 6y – 2z =9". 2nd person says: "I discovered that the maximum value of the directional derivative of the function f at the point P is 5" 3rd person says: "I calculated the derivative of the function f in the direction of vector ū=i +27 + 2k at the point Pas 11". The fourth person, who heard these people's conversations and has no knowledge about the function f, says: "I think that at least one of the three friends made wrong calculations”. So explain the reason of this fourth person's thought in detail with the calculations you need to do. (15)
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ANSWER
case I : Then the annual rector of the plane at point'' is solution The first person say's Right : n =< of of ay Grad f = of (f) = 5 (3, 6,-2) atp. max value of the ditt eential Decivative off' at the of >= 43,6, -27 2 an 22 point "?" = 1of1 = √3)² + (6) ²+ (-2)2 - 9+3 6+4 -√49 = 7+5 > second person is wrong - Derivative of it is the Direction of u= 61+ 2+2 (1,2,2) is at (P). ū =(3, 6, 2> <1,2,27 ST 3+12 - 4 Vi+4+4 @zen 3= 3rd peeson wrong? so : one peeson (1st) is right and the other two (2nd and 3rd) ale or corong case-11 If the ist may be peeson is " WRONG & if possible that the other line might be right (or) one is right one (ory Both are wrong so, In Both the cases at least one peeson is wrong is wrong
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