Column I, Column II `A tx=1,f(x)={logx ,x<1 2x-x^2, xgeq1` , p. is increasing At `x=2,f(x)={x-1,x<2 0,x=2sinx ,x >2` , q. is decreasing At `x=0,f(x)={2x+3,x<0 5,x=0x^2+7,x >0` , r. has point of maxima At `x=0,f(x)={e^(-x)x<0 0,x=0-cosx ,x >0` , s. has point of minima

f(x)=4tanx-tan^2x+tan^3x ,x!=npi+pi/2, (a)is monotonically increasing (b)is monotonically decreasing (c)has a point of maxima (d)has a point of minima

At x=0 the function f(x)=x^3+1 has -

f(x)={cos((pix)/2),x >0, x+a ,xlt=0 Find the values of a if x=0 is a point of maxima.

If f(x)=x^2-3x+1 , then find f(0) .

f(x)={x ,xlt=0 1,x=0,t h e nfin d("lim")_(xvec0)f(x)x^2,x >0ife xi s t s

Column I, Column II (a)sin^(-1)x+x >0, (p) x 0, (q) x in [0,1] (c)tan^(-1)x+x >0, (r) x in [-1,0] (d)cot^(-1)x+x >0, (s) x >0

Statement 1: f(x)=|x-1|+|x-2|+|x-3| has point of minima at x=3. Statement 2: f(x) is non-differentiable at x=3.

Consider the function f:(-oo,oo)to(-oo,oo) defined by f(x)=(x^2-a)/(x^2+a),a >0, which of the following is not true?(a) maximum value of f is not attained even though f is bounded. (b) f(x) is increasing on (0,oo) and has minimum at x=0 (c) f(x) is decreasing on (-oo,0) and has minimum at x=0. (d) f(x) is increasing on (-oo,oo) and has neither a local maximum nor a local minimum at x=0.

Match the following for the type of discontinuity at x=1 in column II for the function in column I. f(x)=1/(x-1) , p. Removable discontinuity f(x)=(x^3-x)/(x^2-1) , q. Non-removable discontinuity f(x)=(|x-1|)/(x-1) , r. Jump of discontinuity f(x)=sin(1/(x-1)) , s. Discontinuity due to vertical asymptote , t. Missing point discontinuity , u. Oscillating discontinuity

Let f(x)={(1+3x)^(1/x), x != 0e^3,x=0 Discuss the continuity of f(x) at (a) x=0, (b) x=1