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Rolle’s theorem is applicable to the and continuous in open interval (a, b) only and having same value at point ‘a’ and ‘b’ b) Functions continuous in closed interval only and having same value at point ‘a’ and ‘b’ c) Functions continuous in closed interval and differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’ d) Monotonically Increasing funtions

Correct Answer: a, b
Statement of Rolle’s Theorem is that, If function f(x) attains same value at point ‘a’ and ‘b’ , and continuous in closed interval and differentiable in open interval (a, b), then there exists a point ‘c’ such that c∈(a,b) and f’(c) = 0.

Mean Value theorem is applicable to the and continuous in open interval (a, b) b) Functions continuous in closed interval only and having same value at point ‘a’ and ‘b’ c) Functions continuous in closed interval and differentiable in open interval (a, b) d) Functions differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’

The mixing vat in a factory receives liquid ingredients through six separate value--- labeled R.S.T.U,Y and Z -- each of which has exactly two settings: open and closed.The mixing vat operator must ensure that each valve is set open or closed according to the following conditions: If T is open, both S and Z must be closed, R and Z cannot both be closed at the same time; If Y is closed, Z must also be closed;S and U cannot both be open at the same time. If the maximum number of valves that can be closed at the same time are closed which of the following must true?

The mixing vat in a factory receives liquid ingredients through six separate value--- labeled R.S.T.U,Y and Z -- each of which has exactly two settings: open and closed.The mixing vat operator must ensure that each valve is set open or closed according to the following conditions: If T is open, both S and Z must be closed, R and Z cannot both be closed at the same time; If Y is closed, Z must also be closed;S and U cannot both be open at the same time. If R is closed and U is open, which of the following must be true?

The mixing vat in a factory receives liquid ingredients through six separate value--- labeled R.S.T.U,Y and Z -- each of which has exactly two settings: open and closed.The mixing vat operator must ensure that each valve is set open or closed according to the following conditions: If T is open, both S and Z must be closed, R and Z cannot both be closed at the same time; If Y is closed, Z must also be closed;S and U cannot both be open at the same time. If Z is open, which of the following must be true?

The mixing vat in a factory receives liquid ingredients through six separate value--- labeled R.S.T.U,Y and Z -- each of which has exactly two settings: open and closed.The mixing vat operator must ensure that each valve is set open or closed according to the following conditions: If T is open, both S and Z must be closed, R and Z cannot both be closed at the same time; If Y is closed, Z must also be closed;S and U cannot both be open at the same time. Which of the following, if given to the mixing vat operator as an instruction, would not determine the setting of any other valve ?

Find the value of ‘a’ if f(x) = ax2+32x+4 is continuous over and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero

Find the value of ‘a’ & ‘b’ if f(x) = ax2 + bx + sin(x) is continuous over and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = π⁄4.

Given the following statements about a function f : R → R , select the right option:
P: If f(x) is continuous at x = x0, then it is differential at x = x0.
Q: If f(x) is continuous at x = x0, then it may not be differentiable at x = x0.
R: If f(x) is differentiable at x = x0, then it is also continuous at x = x0.

Another form of Rolle’s theorem for the continuous condition is _____ b) f is continuous on c) f is continuous on d) f is continuous on

A monotonic function on is either a monotonically increasing or monotonically decreasing function.