
(a)
Whether the statement “The level curves of
(a)

Answer to Problem 1RE
The statement is true.
Explanation of Solution
The given function is,
Let
Take log on both sides.
Here,
Therefore, the statement is true.
(b)
Whether the equation
(b)

Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The equation is
Calculation:
The given equation is
When
The functions are
Therefore, the statement is false.
(c)
Whether the function f satisfies the derivative
(c)

Answer to Problem 1RE
The statement is false.
Explanation of Solution
Let the function f has a continuous partial derivatives of all orders.
Then prove that
For example, assume
Obtain the value of
Take partial derivative of the function f with respect to x and obtain
Thus,
Take partial derivative of the equation (1) with respect to x and obtain
Hence,
Again, take partial derivative for the equation (2) with respect to y and obtain
Therefore,
Obtain the value of
Take partial derivative of the function f with respect to y and obtain
Thus,
Take partial derivative of the equation (1) with respect to y and obtain
Hence,
Again, take partial derivative for the equation (2) with respect to x and obtain
Therefore,
From above, it is concluded that
Thus,
Therefore, the statement is false.
(d)
Whether the gradient
(d)

Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The surface is
Theorem used:
The Gradient and Level Curves:
“Given a function f differentiable at
Description:
The given surface is
By above theorem, it can be concluded that the line tangent to the level curve of f at
Thus, it does not satisfy the given statement. Because, it is given that the gradient
Here,
Therefore, the statement is false.
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