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

[karush]

Does the function satisfy the hypothesis of the mean value theorem
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 $$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

 

[Prove It]

Does the function satisfy the hypothesis of the mean value theorem
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?

 

[karush]

Not sure, the book didn't address where $f'$ is undefined

 

[Prove It]

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...

 

[karush]

OK, thanks.