**Lagrange’s Mean Value Theorem:**

If a function is continuous on the interval and differentiable at all interior points of the interval, there will be, within , at least one point c, , such that .

**Cauchy’s Generalized Mean Value Theorem:**

If and are two functions continuous on an interval and differentiable within it, and does not vanish anywhere inside the interval, there will be, in , a point , , such that .

*Some questions based on the above:*

Problem 1:

Form Lagrange’s formula for the function on the interval .

Problem 2:

Verify the truth of Lagrange’s formula for the function on the interval .

Problem 3:

Applying Lagrange’s theorem, prove the inequalities: (i) (ii) , for . (iii) for . (iv) .

Problem 4:

Write the Cauchy formula for the functions , on the interval and find c.

*More churnings with calculus later!*

Nalin Pithwa.

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