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A function y = 5x2 + 10x is defined over an open interval x = (1, 2). At least at one point in this interval, dy/dx is exactly
- a)20
- b)25
- c)30
- d)35
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A function y = 5x2 + 10x is defined over an open interval x = (1, 2). ...
Understanding the Function
The function given is y = 5x² + 10x. We need to find the derivative dy/dx to analyze its behavior over the interval (1, 2).
Calculating the Derivative
To find dy/dx:
- Differentiate the function:
- dy/dx = d/dx (5x² + 10x)
- dy/dx = 10x + 10
Evaluating the Derivative
Now, we will evaluate the derivative at the endpoints of the interval (1, 2):
- For x = 1:
- dy/dx = 10(1) + 10 = 20
- For x = 2:
- dy/dx = 10(2) + 10 = 30
Applying the Mean Value Theorem
According to the Mean Value Theorem, since the function is continuous and differentiable over the open interval (1, 2), there exists at least one point c in (1, 2) such that:
- dy/dx = (y(2) - y(1)) / (2 - 1)
Calculating y(1) and y(2)
- y(1) = 5(1)² + 10(1) = 15
- y(2) = 5(2)² + 10(2) = 40
Finding the Average Rate of Change
Now, calculate the average rate of change:
- Average Rate = (40 - 15) / (2 - 1) = 25
Conclusion
Since dy/dx varies linearly between the endpoints and includes the average rate of change (25), it confirms that at least one point in the interval has a derivative of exactly 25. Thus, the correct answer is option 'B'.
The function given is y = 5x² + 10x. We need to find the derivative dy/dx to analyze its behavior over the interval (1, 2).
Calculating the Derivative
To find dy/dx:
- Differentiate the function:
- dy/dx = d/dx (5x² + 10x)
- dy/dx = 10x + 10
Evaluating the Derivative
Now, we will evaluate the derivative at the endpoints of the interval (1, 2):
- For x = 1:
- dy/dx = 10(1) + 10 = 20
- For x = 2:
- dy/dx = 10(2) + 10 = 30
Applying the Mean Value Theorem
According to the Mean Value Theorem, since the function is continuous and differentiable over the open interval (1, 2), there exists at least one point c in (1, 2) such that:
- dy/dx = (y(2) - y(1)) / (2 - 1)
Calculating y(1) and y(2)
- y(1) = 5(1)² + 10(1) = 15
- y(2) = 5(2)² + 10(2) = 40
Finding the Average Rate of Change
Now, calculate the average rate of change:
- Average Rate = (40 - 15) / (2 - 1) = 25
Conclusion
Since dy/dx varies linearly between the endpoints and includes the average rate of change (25), it confirms that at least one point in the interval has a derivative of exactly 25. Thus, the correct answer is option 'B'.
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Community Answer
A function y = 5x2 + 10x is defined over an open interval x = (1, 2). ...
y = f(x) = 5x2 + 10x in the internal x = (1, 2)
Since, the function y is continuous in the interval (1, 2), as well as is differentiable at each point so, from Lagrange mean value theorem there exist at least a point where:
Here, we have
a = 1, b = 2
So, for x = a = 1, we obtain:
y = f(a) = f(1) = 5(1)2 + 10(1) = 15
and for x = b = 2
y = f(b) = f(2) = 5(2)2 + 10(2) = 40
Therefore:
Since, the function y is continuous in the interval (1, 2), as well as is differentiable at each point so, from Lagrange mean value theorem there exist at least a point where:
Here, we have
a = 1, b = 2
So, for x = a = 1, we obtain:
y = f(a) = f(1) = 5(1)2 + 10(1) = 15
and for x = b = 2
y = f(b) = f(2) = 5(2)2 + 10(2) = 40
Therefore:
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A function y = 5x2 + 10x is defined over an open interval x = (1, 2). At least at one point in this interval,dy/dx is exactlya)20b)25c)30d)35Correct answer is option 'B'. Can you explain this answer?
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