Is the given function differentiable at the endpoints on the interval [0,5]?

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In summary, the function $f(x)=\sqrt{x(5-x)}$ on the interval $[0,5]$ is continuous at all points but not differentiable at the endpoints due to vertical slopes. It does not satisfy the hypothesis of the mean value theorem because the derivative is not defined at the endpoints. The mean value theorem states that the gradient of any chord on a continuous and smooth function is equal to the gradient at some point in between, but this does not apply to $f$ on $[0,5]$ because $f'$ is undefined at the endpoints.
  • #1
karush
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Does the function satisfy the hypothesis of the mean value theorm
On the given interval, give reasons for your answer

$$f\left(x\right)=\sqrt{x\left(5-x\right)},\ \ \left[0,5\right]$$

The graph of this is the top half of a circle

My answer was
It is continuous at all points on the interval but not differential at the endpoints due to vertical slopes.

This answer was not correct.?

MVT $$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$
 
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  • #2
karush said:
Does the function satisfy the hypothesis of the mean value theorm
On the given interval, give reasons for your answer

$$f\left(x\right)=\sqrt{x\left(5-x\right)},\ \ \left[0,5\right]$$

The graph of this is the top half of a circle

My answer was
It is continuous at all points on the interval but not differential at the endpoints due to vertical slopes.

This answer was not correct.?

MVT $$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

What is your hypothesis about the MVT?
 
  • #3
Not sure, the book didn't address where $f'$ is undefined
 
  • #4
karush said:
Not sure, the book didn't address where $f'$ is undefined

I don't think you understand the mean value theorem very well. It states that if your function is continuous everywhere and smooth at all points except the endpoints, then the gradient of any chord on that function is the same as the gradient of the function at some point in between. So your derivative doesn't need to be defined at the endpoints...
 
  • #5
OK, thanks.
 

FAQ: Is the given function differentiable at the endpoints on the interval [0,5]?

What is the N22.24 Mean value theorem?

The N22.24 Mean value theorem is a mathematical theorem that relates the average rate of change of a function to its instantaneous rate of change at a specific point. It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) where the slope of the tangent line at c is equal to the average rate of change of the function on the closed interval [a, b].

How is the N22.24 Mean value theorem applied in calculus?

In calculus, the N22.24 Mean value theorem is used to prove the existence of local extrema (maximum or minimum) of a function on a closed interval. It is also used to determine the value of integrals, as it provides a way to find the average value of a function over an interval.

What is the difference between N22.24 Mean value theorem and Rolle's theorem?

N22.24 Mean value theorem is a generalization of Rolle's theorem. While Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval, with equal values at the endpoints of the interval, then there exists a point where the derivative of the function is equal to 0. The N22.24 Mean value theorem does not require the function to have equal values at the endpoints, but instead states that there exists a point where the derivative is equal to the average rate of change of the function on the closed interval.

What is the significance of the N22.24 Mean value theorem?

The N22.24 Mean value theorem is significant in calculus as it provides a powerful tool for analyzing the behavior of functions. It allows us to determine the existence of local extrema and average values of a function over an interval. It also serves as a key step in the proof of the Fundamental Theorem of Calculus.

Can the N22.24 Mean value theorem be applied to all types of functions?

Yes, the N22.24 Mean value theorem can be applied to all continuous and differentiable functions on a closed interval. However, it may not always be possible to find a point where the derivative is equal to the average rate of change of the function, especially for more complex functions. In such cases, the theorem can still be used to prove the existence of local extrema and average values, even if the point itself cannot be determined.

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