A differentiable function `f(x)` has a relative minimum at `x=0.` Then the function `f=f(x)+a x+b` has a relative minimum at `x=0` for all `a` and all`b` (b) all `b` if `a=0` all `b >0` (d) all `a >0`

The function f(x)=x/2+2/x has a local minimum at x=2 (b) x=-2 x=0 (d) x=1

Prove that for all a,b in the function f(X) =3x^(4)-4x^(3)+6x^(2)+ax+b has exactly one extremium.

If f:RR-> RR is a differentiable function such that f(x) > 2f(x) for all x in RR and f(0)=1, then

The function f(x)=sin^(-1)(cosx) is (a) discontinuous at x=0 (b) continuous at x=0 (c) differentiable at x=0 (d) none of these

Let f(x+y)=f(x)+f(y) for all xa n dydot If the function f(x) is continuous at x=0, show that f(x) is continuous for all xdot

Let f: R->R be a differentiable function for all values of x and has the property that f(x)a n df^(prime)(x) has opposite signs for all value of xdot Then, (a) f(x) is an increasing function (b) f(x) is an decreasing function (c) f^2(x) is an decreasing function (d) |f(x)| is an increasing function

If a continous function of defined on the real line R, assumes positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. The positive value of k for which ke^(x)-x=0 has only root is

If a continuous function of defined on the real line R, assumes positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. For k gt 0, the set of all values of k for which ke^(x)-x=0 has two distinct, roots, is

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x) le 2 for all x. Justify your answer.

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x)le2 for all x. Justify your answer.