Question

let f is differentable
(a,b) then f
is
on (a,b) if
f'(x) <0 tx € (a,b)​

Answer 1

$\huge\boxed{\fcolorbox{violet}{violet}{Answer}}$

Assume that f:[a,b]

→R is a function.

Prove the following statement :

If f is differentiable on [a,b] and f′ is monotonic on [a,b], then f′ is continuous on [a,b].

Note 1 : We've learned a theorem in the class which says :

If I is an open interval and f:I

→R is monotonic, then f is continuous on I.

So, here i can say that f′ is continuous on (a,b). But my problem is how to prove that f′ is also continuous at the end points a,b.

Answer 2

$\huge{\fcolorbox{pink}{violet}{Answer }}$

Assume that f:[a,b]

→R is a function.

Prove the following statement :

If f is differentiable on [a,b] and f′ is monotonic on [a,b], then f′ is continuous on [a,b].

Note 1 : We've learned a theorem in the class which says :

If I is an open interval and f:I

→R is monotonic, then f is continuous on I.

So, here i can say that f′ is continuous on (a,b). But my problem is how to prove that f′ is also continuous at the end points a,b.