Practice Questions on Differentiability and Answers

**Questions on Differentiability and Answers**

**Q1. Let X = (0,1)∪(2,3) be an open set in R. Let f be a continuous function on X such that the derivative f'(x) = 0 for all x. Then the range of f has**

(a) Uncountable number of points

(b) Count ably infinite number of points

(c) At most 2 points

(d) At most 1 point

**Q2. If f(x) is real valued function defined on [0,∞] such that f(0) = 0 and f ^{“}(x) > 0 for all x, then the function h(x) = ^{f(x)}/_{x} is**

(a) Increasing in [0,∞]

(b) Decreasing in [0,1]

(c) Increasing in [0,1] and decreasing in [1,∞]

(d) Decreasing in [0,1] and increasing in [1,∞]

**Q3. Let f:R ****→ ****R be a continuous function and f(x+1) = f(x) for all x****∈****R. Then**

(a) f is bounded above, but not bounded below.

(b) f is bounded above and below but may not attain its bounds

(c) f is bounded above and below and attains its bounds

(d) f is uniformly comtituous

**Q4. Decide which of the following functions are Uniformly continuous on (0,1).**

(a) f(x) = e^{x}

(b) f(x) = x

(c) f(x) = tan(^{πx}/_{2})

(d) f(x) = sin(x)

**Q5. Which of the following statements is (are) on the interval (0,****π/2) ?**

(a) cosx < cos(sin x)

(b) tanx < x

(c) √(1+x) < 1 + x/_{2} – x^{2}/_{8}

(d) (1-x^{2})/_{2} < In(2+x)

**Q6. Let f:R → R be a differentiable function with f(0)=0 If for all x∈R, l < f ^{1}(x) < 2, then Which one of the following statements is true on (0,∞) ?**

(a) f is unbounded

(b) f is increasing and bounded

(c) f has at least one zero

(d) f is periodic

**Q7. Let f(x) = ^{1}/_{(x+|x|)} + ^{1}/_{(x+|x-1|)} for all x∈[-1,1]. Then which one of the following is TRUE ?**

(a) Maximum value of f(x) is ^{3}/_{2}

(b) Minimum value of f(x) is ^{1}/_{2}

(c) Maximum of f(x) occurs at x=^{1}/_{2}

(d) Minimum of f(x) occurs at x=1

**Q8. Consider the function f(x) = |cosx|+|sin(2-x)| . At which of the following points is f not differentiable?**

(a) {(2n+1)^{π}/_{2} : n∈Z}

(b) {nπ : n∈Z}

(c) {nπ+2 : n∈Z}

(d) {^{nπ}/_{2} : n∈Z}

**Q9. Let F = {f:R → R: |f(x)-f(y)| ≤ K|(x-y)| ^{a}} For all x, y∈R and for some a > 0 and some K > 0.Which of the following is/are true?**

(a) Every f∈F is continuous

(b) Every f∈F is uniformly continuous

(c) Every differentiable function f is in F

(d) Every f∈F is differentiable

**Q10. Let f:R → R be defined by f(x) = x ^{2}sin(^{1}/x^{2}), x ≠ 0, f(0) = 0**

(a) f is differentiable everywhere with continuous derivative

(b) f is differentiable everywhere other than the point 1

(c) f is nowhere differentiable

(d) f is differentiable but its derivative is not continuous

**Q11. Let f:(0,1) → R be continuous. Suppose that** **|f(x) – f(y)| ≤ |cosx – cosy| for all x,y ∈ (0,1). Then**

(a) f is discontinuous at least at one point in (0,1).

(b) f is continuous everywhere on (0,1), but not uniformly continuous on (0,1).

(c) f is uniformly continuous on (0,1).

(d) lim f(x) exists, where x → 0

**Q12. Let f(x) = |sin****πx| ,**** x****∈****R.Then**

(a) f is contimuous nowhere

(b) f is continuous everywhere and differentiable nowhere

(c) f is continuous everywhere and differentiable everywhere except at integral value of x

(d) f is differentiable everywhere

**Q13. The function f(x) = 1-|1-x| on R is**

(a) Not continuous

(b) Continuous but not differentiable

(c) Differentiable but not continuous

(d) Differentiable at only one point

**Q14. f:[0,1] ****→ ****R is a function. Which of the following is possible?**

(a) f is differentiable but not continuous

(b) S is unbounded and continuous

(c)S is differentiable but its derivative is not continuous

(d) f is continuous but not integrable

**Q15. Let f = R ****→ ****R be differentiable. Which of these will follow from mean value theorem**

(a) For all a, b in R, if a<c<b then ^{[f(b) – f(a)]}/_{(b-a)}**= f ‘(c)**

(b) For some a, b in R and for all c in (a, b), ^{[f(b) – f(a)]}/_{(b-a)} = f ‘(c)

(c) For all a, b in R there is some c in (a, b) such that** ^{[f(b) – f(a)]}/_{(b-a)} = f ‘(c)**

(d) For all a, b in R there is a unique element c in (a, b) then the

^{[f(b) – f(a)]}/_{(b-a)}= f ‘(c)NOTE:- If you need anything else like e-books, video lectures, syllabus, etc. regarding your Preparation / Examination then do 📌 mention it in the Comment Section below. |

*Answer Key*

*Answer Key*

01. | (c) | 06. | (a) | 11. | (c), (d) |

02. | (a) | 07. | (a) | 12. | (c) |

03. | (c), (d) | 08. | (a), (c) | 13. | (b) |

04. | (a), (b), (d) | 09. | (a), (b) | 14. | (c) |

05. | (a), (d) | 10. | (d) | 15. | (c) |