Question

Let \( f(x,y,z)=xy+yz+xz \). If a point \( (0,0,\lambda) \) lies on the tangent plane to the surface \( f(x,y,z)=3 \) at the point \( (1,1,1) \), then the value of \( \lambda \) is __________.

Hint:

For a level surface \( f(x,y,z)=c \), the tangent plane at a point is obtained using the gradient vector evaluated at that point.

Answer 1

Correct Answer: 3

The tangent plane to a surface \( f(x,y,z)=3 \) at point \( (1,1,1) \) is given by:
\[ \nabla f(1,1,1) \cdot (x-1,\ y-1,\ z-1)=0 \]
First compute the gradient:
\[ \frac{\partial f}{\partial x}=y+z,\quad \frac{\partial f}{\partial y}=x+z,\quad \frac{\partial f}{\partial z}=x+y \]
At the point \( (1,1,1) \):
\[ \nabla f(1,1,1)=(2,\ 2,\ 2) \]
Tangent plane equation becomes:
\[ 2(x-1)+2(y-1)+2(z-1)=0 \]
Divide by 2:
\[ (x-1)+(y-1)+(z-1)=0 \]
\[ x+y+z=3 \]
Since \( (0,0,\lambda) \) lies on the plane:
\[ 0+0+\lambda = 3 \]
Thus,
\[ \lambda = 3 \]

Final Answer: 3