# Questions on Differentiability and Answers

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 xR. 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) = ex
(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 – x2/8
(d) (1-x2)/2 < In(2+x)

Q6. Let f:R → R be a differentiable function with f(0)=0 If for all x∈R, l < f1(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) {/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) = x2sin(1/x2), 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) = |sinπx| , xR.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)