Suppose that f satisfies the hypotheses of the Mean Value Th

Suppose that f satisfies the hypotheses of the Mean Value Theorem on [a, b]. If lim f\'(x) = A, then f\' _+(a) exists and is equal to A. [A may be finite or infinite.]

Solution

if f satisfied hypotheses of mean value theorem

it means that f is continuous and derivable for all the points between [a,b]

now f\'(a) = lim x -> a f(x) - f(a) / (x-a) = A

lim h -> 0 f(a+h) - f(a) / h = A

which exist and is equal to A