Differentiability (Theorems)

If we know that f is 3 times differentiable then
(choose all)

• f is continuous

• f' is continuous

• f'' is continuous

• f''' is continuous

Which of these is NOT a hypothesis of the Mean Value Theorem (MVT)?

• Function is continuous on a closed, bounded interval

• Function is differentiable on the interior of a bounded interval

• Function gets same values at the ends of an interval

BONUS QUESTION:
Why in MVT there is a condition of continuity (as there is already a condition of differentiability)?

The function, F, above satisfies the conclusion of Rolle's Theorem in the interval [a,b] because:

I. F is continuous on [a,b]

II. F is differentiable on (a,b)

III. F(a) = F(b) = 0

• All 3 statements are true

• I is not true

• II is not true

• None of the statements are true

• YES
• NO, no continuity

• NO, no differentiability

• NO, no continuity and no differentiability

• I only

• II only

• III only

• II and III only

• I, II, and III

If f is continuous on [-1,1], f (− 1)=4 and f(1)=−2 , then there are a,b: a≥f(x)≥b

• Extreme value theorem

• Mean Value Theorem

• Intermediate value theorem

• Rolle's theorem

• I only

• II only

• I and III only

• I, II, III

• The graph of f is concave up.

• The graph of f has at least two points of inflection.

• f is increasing.

• f has at least two relative extrema.

• M

• N

• P

• Q