QQuestionMathematics
QuestionMathematics
Find the point(s) on the surface at which the tangent plane is horizontal.
z= 9−x 2 −y 2 + 8y
(x,y,z)=(…)
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Answer
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Step 1:I'll solve this step by step, carefully following the LaTeX formatting guidelines:
Step 2:: Find the partial derivatives
\frac{\partial z}{\partial y} = -2y + 8
To find points where the tangent plane is horizontal, we need to find where the partial derivatives with respect to x and y are both zero. Partial derivative with respect to x: Partial derivative with respect to y:
Step 3:: Set partial derivatives to zero
-2y + 8 = 0
Set both partial derivatives equal to zero to find the critical points:
Step 4:: Solve for x and y
y = 4
From the first equation: From the second equation:
Step 5:: Calculate the z-coordinate
z = 25
Substitute x = 0 and y = 4 into the original equation:
Final Answer
The point where the tangent plane is horizontal is (0, 4, 25).
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