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Let f :R-+R be a continuous function with, , period p>0.Then g(x)= | fdr isa, , , , , pifferentiability, , , , vr, Let f(x) =sinx-4+5 and, , 2, glx)=cosx-1+ 5 for xR. Which of the, , following statements are correct?, , , , , , , , , , , , , , , , , (a.) Constant function, (b.) Continuous function (a.) f(x)20 forall x>0, (c.) Continuous function but — not Se . ., differentiable (b.) g is an increasing function on [0, =), (d.) Neither continuous nor differentiable (c.) g is an decreasing function on [0, =), ae is A be the set of rational numbers in the (4) J is a decreasing function on [0, «), open interval (0, 7) and f:4+R bea :, uniformly continuous function. Which of the 1 bas =[0.]ek - For xeR , let, following ere true? SSG) = aistlond)=int (be ys € 1} Then, (a) f is bounded ~~, 7s on We g(x) is discontinuous somewhere on R, (b) f is necessarily a constant function ~~ <1 . 4, ; Qr (b.) e(x) is continuous on R but not, (c.) f is differentiable on (0, 7) YN continuously differentiable exactly at, . . Yo x=0, (4) f is differentiable at all the, pomts in (0, 7) os eal (c.) g(x) is continuous on R but not, 11. Lat f be a twice differentiable function — continuously differentiable exactly at, =0 and x=, BR. Give that /"(s)> Oforall xe.” ys zad-antls=s, So (d) e(x)is differentiable on R (2) (f(z) =Ohas exactly two solution on Rg, 15. Consider the function, (x)= iti ion i (b) fl2)=0 has postive soltion if 2) =cox()x-$) +sin( 2) +J2+108 - (+4, S(9)=6 and £'(0)=0 ‘At which Of the following point is f not, (c.) flz)=0 hes no positive solution if differentiable?, s(0)=0 end £(0)>0 (a) x=5, (4) fla)=0 tas no positive solution if (b) x=3, flo) =0 and $(0)<0 (c.) x=-10, 12. Suppose f-R—+R is 2 AGifierentizble (d) x=0, fonction Then which of the following 16. Define :ROR by, statements are necessarily true? yy { 2 ift<0 7. which of th, la) sere! forall xeR, then f has Matin, at keast one fixed point. following statements are correct?, , (b)If f bas a unique fixed point, then, S()sr<\ forall xeR, , (c)If f has 2 unique fixed point, then, Sa)zr>-\ forall reR, , (4.) None of these, , (a.) "(x)=2 forall xeR, , , , , , (b.) £7 (0) does not exist, (c.) JS (x) exists for each x#0

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19., , 20., , , , , , , , Consider the function 21., , J(x)=|cosx|+|sin(2-x)| At which of the, following points is_f not differentiable?, , (a) [(2n+1}8-ne2|, , (.) {nz:nez}, , (C.) {nx +2:neZ} 2, , (4) (= neZ, , Let, F={f:RR:[f(x)- fv) SK|(z-y}, , , , Coe LL, , , , The function f(x) =|xi+3 is, , (a.) Continuous as well as differentiable on, R, , (b.) Continuous on R but not differentiable, anywhere on R, , (c.) Continuous on R and differentiable at all, but finitely many points on R, , (d.) Non continuous on R, , Let f(x) =|sinzxi, x¢R Then, , (a.) f is continuous nowhere, , (b.) f is continuous everywhere and, differentiable nowhere, , N@)- sf is~continuous everywhere and, , For all x, y<R and for some a >0 and some _™~ differentiable everywhere except at, X >0.Which of the following is/are true? an, integral value of x, (a.) Every fe Fiscontinuous “~~ * \(d) Fis differentiable everywhere, (b.) Every f¢ Fis uniformly continuous OT The function f(x)=1-l1-x| on R is, (c.) Every differentiable function f isin K-~_\~— _ @) Not continuous, - ; Nw (b.) Continuous but not differentiable, (d) Every fc F is differentiable LC (c.) Differentiable but not continuous, Let f:R+R be defined So (d.) Differentiable at only one point, atevie Levee, stile \ ~ 2%. £:[0, 1]+R is a function. Which of the, r a following is possible?, (a) f is differentiable everywhere with (a.) f is differentiable but not continuous, Continuous derivative (b.) £ is unbounded and continuous, (b.) £ is differentiable everywhere other (c.) f is differentiable but its derivative is, than the point I not continuous, (c.) f is nowhere differentiable (4) F is continuous but not integrable, (4) f is differentiable but its derivative is 25. Let f-R-R be differentiable. Which of, , not continuous, Let f :(0,1)-+R be continuous. Suppose that, \f(2)-1(y)|$ cos x-cos y| 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 uniform :, , these will follow from mean value theorem?, (a) For all a,b in R, if a<c<b then, ‘b)f(b) L@ _ py), , b-a, (b.) For some a,b in R and for all ¢ in, S(b)- f(a) _, (a, 6), a =f), , (c.) For all a, in R, there is some.c in, , (a, 6) such that LO). 546, , (d.) For all a,5 in R, there is a unique, , element cin (a,b) then the, b) ~, SC I=L - F, , a, , , , Ph: (O11)-26557527, Cell: 9999193434 & 9999161734, ESRRRAITED _, wwwdipsecademy.com

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L, An t$0 9001 ; ce, I 30., 26. The function (,) « ssia( + ) x40 i,, 0 x=0, , 27., , , , , (a.) Differentiable at 0 but not continuous, (b.) Has 2™ derivative at the origin, , (c.) Continuous at the origin but not, differentiable, , (d.) Neither continuous nor differentiable at, the origin, , 3., , The set of points where f(x) =|sinx| is not, differentiable is, , (a.) Empty, (b.) {0}, (c.) {kx :keZ} \ —, , (4) [fe :tez}, , Let £:(0,2)9R be cies, , r(s)-{ x’ Ifx is rational ve, , 2x-1 Ifx is irrational, (a.) f is differentiable exactly at one point, (b.) f is differentiable exactly at two points, , (c.) f is not differentiable at any point in, (0,2), , (d.) f is differentiable at every point in, 0.2) =, , SRR is such that, eo, J(}) is in, (a) (5,6), (b.) [-5,5], , (c.) (-2,-$)0(5,20), , f(0)=0 and, , <5 for all x, We can conclude that, , , , (4) [-4,4], , , , Ss, , , , Let f:R->R be a differentiable function, , such that sup,<x |/“(x)| <%- Then, , (a.) f maps @ bounded sequence to a, bounded sequence., , (b.) f maps a Cauchy sequence to a, Cauchy sequence., , (c.) f maps a convergent sequence to a, convergent sequence,, , (d.) Jf is uniformly continuous., , Let f:R>R be a twice continuously, _ differentiable function, with, O=1{01= £(0)= 0. Then, , wo Sf’ is the: Zero function., , Sh f° (0) is zero., , (c.) f*(x)=0 for some x€(0,1)., , (d.) f” never vanishes,, , Let f:[0,©)-+[0,0) be a continuous, , function. Which of the following is correct?, , (a.) There is x) €[0, 0) such that f (x9) = x9, , (b.) If f(x) <M for all xe[0,) for some, M>0, then there exists x ¢[0, «), such that f (x9) = x9., , (c.) If f has a fixed point, then it must be, unique., , (d.) f does not have a fixed point unless it, is differentiable on (0, 0)., , Take the closed interval [0,1] and open, interval (1/3,2/3). Let k =[0,1]\(1/3,2/3)., For xe[0,1] define f(x)=d(x,K) where, S(x)=(x,K)= inf {|x- yy eK}. Then, , (a) f:[0,1]>R is differentiable at all, , points of (0,1)., , (&) f:[0,1]>R is not differentiable at, , 1/3 and 2/3., , (c.) f:[0,1]->R is not differentiable at, , 1/2,

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~wewwewrwrwerr,, , 35,, , 36., , , , 34., , Let X =(0,1)U4(2,3) be an open set in R. 38., Let be a continuous function on X such, that the derivative f'(x)=0 for all x. Then, the Tange of f has, (@.) Uncountable number of points, (b.) Count ably infinite number of points, (c,.) At most 2 points, (d.) At most 1 point, If S (x) is real valued function defined on 39., [0,0] such that £(0)=0 and f"(x)>0, £0) ,, , for all x, then the function A(x) =, , WN, (a.) Increasing in [0,00] TSO, (b.) Decreasing in [0,1] XS), (c.) Increasing in [o, 1] and decreasin, sSNA, yo, , [1.~], (4.) Decreasing in [0,1] and weragg, ae, [1], Let f:R-—> Rbe a continuous function and, S(x+1)= f(x) for all xe R. Then, , (a.) fis bounded above, but not bounded, below., , (b.) f is bounded above and below, but, may not attain its bounds., , (c.) fis bounded above and below and Sf, attains its bounds. 41., , (4) / is uniformly continuous, Decide which of the following functions are, uniformly continuous on. (0,1) ., , (a) £(*)=, OUNIE, , , , , , , , , , , PT Come LL, , Which of the following statements is (are) on, , the interval (05), , (a.) cosx < cos(sin.x), , (b.) tanx<x, (c.) Vive <t+5-—, , , , (d.) <In(2+x), , Let f:R-—R be au differentiable function, , with _/(0)=0 If for all xeR I< f'(x)<2,, “jhen Which one of the following statements, , Nig true on (0,20) 2,, Ta.) fis unbounded, , (b.) / is increasing and bounded, (c.) f has at least one zero, (d.) f is periodic, , 1, ie f(2)= aaa 1+{x-1|, , xe[-1,1]. Then which one of the following, is TRUE?, , , , all, , , , (a.) Maximum value of f(x) is 3, (b.) Minimum value of f(x) is ;, , (c.) Maximum of f(x) occurs at x5, , (d.) Minimum of f(x) occurs at x=1, Let f:R — Rbe defined by, , sinx, — if x#0, 2 ee . Then, , 1 if x=0, , S(x)=, , (a.) f is not continuous, , (b:) / is continuous but not differentiable, (c.) fis differentiable, , (d.) 7 is not bounded, , , , Weber ca S27, Cut SPPUESANG & 9D96TTI4, ASRRRGTD mm