Differentiation

To analyze the function and determine its extremum:

1. Compute the Derivative:
We first find the derivative of :

2. Identify Critical Points:
The derivative is undefined when the denominator becomes zero:


Thus, is a critical point where the derivative does not exist.

3. Determine Extremum at Critical Point:
Evaluating the function at :


Since the function has a well-defined value here while its derivative is undefined, this represents a point of extremum.

4. Verify Minimum Value:
For all :


Therefore, is indeed the minimum value of the function.

Final Answer:
The function has its minimum value at , which is .

To solve the problem, we need to find the derivative of the curve at the point .

1. Differentiate the Given Equation:
We start by differentiating both sides of with respect to :

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2. Apply Product Rule to :
Using the product rule for , we get:

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3. Collect Like Terms:
Gather all terms on one side:

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4. Factor Out :
Factor the left-hand side to solve for :

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5. Solve for the Derivative:
Isolate to find the general expression for the derivative:

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6. Evaluate at the Given Point :
Substitute and into the derivative:

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Final Answer:
The derivative of the curve at the point is .

The given differential equation is: The order of the differential equation is determined by the highest order of the derivative.
In this case, the highest order derivative is , so the order is 5.
The degree of a differential equation is the exponent of the highest order derivative, provided that the equation is free from any fractional powers or irrational expressions in derivatives.
In this case, the highest order derivative is , and its exponent is 1 (since it is not raised to any power).
So, the degree is 1. Thus, the sum of the order and degree is:
Thus, the correct answer is .

To find the turning point, we need to first calculate the derivative of the function. The function given is: We can apply the quotient rule for differentiation: Now, differentiate the numerator and denominator carefully, and substitute to check the nature of the turning point at that point. By analyzing the second derivative or using test points around , we can conclude that the turning point at is a maximum.
Thus, the correct answer is (D).