Rolle's theorem cannot be applied t the function f(x)= ln(x+2) on the interval [-1,2] because
a) f is not differentiable on the interval [-1,2]
b) f(-1)≠ f(2)
c) All of these
d) Rolle's theorem can be applied to f(x)= ln(x+2) on the interval [-1,2]
<<Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the is zero...>>
so the function is ln(1)<ln(4). There are not two points which have equal values.
f'(ln(x+2))=1/(x+2) which is postive slope over the domain of x.
so a) is not true
b) is almost right (but the two points don't have to be endpoints of the domain), but in fact, there are no two points existing.
c. nope
d) yep. Best answer.
To answer this question, let's first understand what Rolle's theorem states. Rolle's theorem is a theorem in calculus that states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are the same, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.
Now, let's apply this to the given function f(x) = ln(x + 2) on the interval [-1, 2].
a) The function f(x) = ln(x + 2) is continuous on the interval [-1, 2] since the natural logarithm function is continuous on its domain. However, in order to apply Rolle's theorem, the function needs to be differentiable on the open interval (-1, 2).
The derivative of f(x) = ln(x + 2) is 1 / (x + 2). However, this derivative is not defined for x = -2, which is within the interval (-1, 2). Therefore, f(x) = ln(x + 2) is not differentiable on the interval (-1, 2). So, option a) is correct.
b) The condition that f(-1) ≠ f(2) is not necessary for the application of Rolle's theorem. Rolle's theorem only requires that the function is continuous on the closed interval [-1, 2] and differentiable on the open interval (-1, 2). So, option b) is incorrect.
c) Since option a) is correct, option c) is incorrect.
d) Since f(x) = ln(x + 2) is not differentiable on the open interval (-1, 2), option d) is incorrect.
Therefore, the correct answer is option a) "f is not differentiable on the interval [-1,2]".