Question

Decide whether Rolle's Theorem can be applied to f(x) = x2 + 5 on the interval [0, 3]. If Rolle's Theorem can be applied, find all value(s) of c in the interval such that f ′(c) = 0.
Rolle's Theorem can be applied; c = negative one half
Rolle's Theorem can be applied; c = 0, 3
Rolle's Theorem cannot be applied because f ′(c) ≠ 0
Rolle's Theorem cannot be applied because f(0) ≠ f(3)

Answer 1

F(x)= x² + 5, is just a parabola shfited upwards by 5 units, so, is a smooth graph and no abrupt edges, so from 0 to 3, is indeed differentiable and continuous.  So Rolle's theorem applies, let's check for "c" by simply setting its variable to 0, bear in mind that, looking for "c" in this context, is really just looking for a critical point, since we're just looking where f'(c) = 0, and is a horizontal tangent line.

$\bf f(x)=x^2+5\implies \cfrac{df}{dx}=2x\implies 0=2x\implies \boxed{0=x}\leftarrow c$