Indicate whether the following statements are True or False. If a statement is t
ID: 2887713 • Letter: I
Question
Indicate whether the following statements are True or False. If a statement is true, explain how you know it is true. If it is false, give a counterexample and explain why it is a counterexample. A graph with an explanation in words can be used as a counterexample.
(a) If a function, f(x), is continuous at x = ?2, then it is also differentiable at x = ?2.
(b) If limx?4 f(x) = 5, then f(x) is continuous at x = 4.
(c) If f '(x) = g 0 (x), then f(x) = g(x).
(d) If f ''(3) = 0, then f has an inflection point at x = 3.
Explanation / Answer
Solution :- a) Given If a function, f(x), is continuous at x = ?2, then it is also differentiable at x = ?2.
Since continuous function need not be differentiable . that is a continuous function may or may not be differentiable at a point x=a .for example consider the example of f(x)=|x| { absolute (x) }. This function is continuous but not differentiable at x = 0. {False}
(b) If limx?4 f(x) = 5, then f(x) is continuous at x = 4.
A function is said to be continuous if limx?a f(x) = f(a) = L
if f(4) =5 then f(x) is continuous at x = 4. otherwise not.
d) If f ''(3) = 0, then f has an inflection point at x = 3 (True)
Please comment it ASAP and what is C