Let `f(1)=-2a n df^(prime)(x)geq4. 2for1lt=xlt=6.` The smallest possible value of `f(6)` is 9 (b) 12 (c) 15 (d) 19

The maximum value of f(x)=x/(1+4x+x^2) is -1/4 (b) -1/3 (c) -1/6 (d) 1/6

Let f be differentiable for all x , If f(1)=-2a n df^(prime)(x)geq2 for all x in [1,6], then find the range of values of f(6)dot

Let f(x)=cospix+10x+3x^2+x^3,-2lt=xlt=3. The absolute minimum value of f(x) is 0 (b) -15 (c) 3-2pi none of these

Let f(x) be a polynomial of degree 3 such that f(3)=1,f^(prime)(3)=-1,f^('')(3)=0,a n df^(''')(3)=12. Then the value of f^(prime)(1) is (a) 12 (b) 23 (c) -13 (d) none of these

Let f(x)a n dg(x) be differentiable for 0lt=xlt=1, such that f(0),g(0),f(1)=6. Let there exists real number c in (0,1) such taht f^(prime)(c)=2g^(prime)(c)dot Then the value of g(1) must be 1 (b) 3 (c) -2 (d) -1

Suppose that f is differentiable for all x and that f^(prime)(x)lt=2fora l lxdot If f(1)=2a n df(4)=8,t h e nf(2) has the value equal to 3 (b) 4 (c) 6 (d) 8

Let f: R->R be such that f(1)=3a n df^(prime)(1)=6. Then lim_(x->0)((f(1+x))/(f(1)))^(1//x)= (a) 1 (b) e^(1/2) (c) e^2 (d) e^3

Let f(x) be continuous functions f: RvecR satisfying f(0)=1a n df(2x)-f(x)=xdot Then the value of f(3) is 2 b. 3 c. 4 d. 5

If ("lim")_(tvecx)(e^tf(x)-e^xf(t))/((t-x)(f(x))^2)=2a n df(0)=1/2, then find the value of f^(prime)(0)dot a) 4 (b) 2 (c) 0 (d) 1

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt