Question

Consider the following two statements and choose the correct answer.

a) If f(x)f(x) is continuous at x = a, then f(x)f(x) is differentiable at x = a.
b) For every fixed constant c, the (natural) domain of f(x) = 3x + ln(x-c) is [c,[infinity]).

Answer 1

Statement a is false: a function can be continuous at a point but not differentiable at that point. Statement b is true: the domain of f(x) = 3x + ln(x-c) is [c, ∞).

Step-by-step explanation:

a) If f(x)f(x) is continuous at x = a, then f(x)f(x) is differentiable at x = a:

This statement is false. A function can be continuous at a point but not differentiable at that point. For example, the function f(x) = |x| is continuous at x = 0, but it is not differentiable at x = 0.

b) For every fixed constant c, the (natural) domain of f(x) = 3x + ln(x-c) is [c,[infinity]]:

This statement is true. The natural logarithm function ln(x) is defined for x > 0, so the domain of f(x) = 3x + ln(x-c) is all values of x such that x > c. In interval notation, the domain is [c, ∞).