Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

If f (x) = |x|³, show that f″(x) exists for all real x and find it.

Using mathematical induction prove that for all positive integers n.

Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

If , prove that