Using Rolles theorem, prove that there is at least one root in `(45^(1/(100)),46)` of the equation. `P(x)=51 x^(101)-2323(x)^(100)-45 x+1035=0.`

Solve the equation : x^4-14x^2+45=0

Using intermediate value theorem, prove that there exists a number x such that x^(2005)+1/(1+sin^2x)=2005.

If 2a+3b+6c=0, then prove that at least one root of the equation a x^2+b x+c=0 lies in the interval (0,1).

Solve the eqation : x^(4)-14x^(2)+45=0

Find the condition if the equation 3x^2+4a x+b=0 has at least one root in (0,1)dot

One root of the equation (12x -1)(6x -1)(4x-1)(3x-1)=5 is

Find the roots of the equation 100x^2-20x+1

Using the Rolle's theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions : f(x)=x^(2)-x,x in[0,1]

Using the Rolle's theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions : f(x)=(x^(2)-2x)/(x+2),x in[-1,6]