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

Using mean value theorem,prove that tan x>x for all x(0,(pi)/(2))

Use the mean value theorem to prove e^(x)>=1+x AA x in R

Using Lagranges mean value theorem,show that sin(:x for x:)0.

Let f(x)=4x^3-3x^2-2x+1, use Rolles theorem to prove that there exist c ,\ 0

Use the Intermediate Value Theorem to show that the polynomial function f(x) = x^(3) + 2x-1 has a zero in the interval [0,1].

Using binomial theorem expand (x^(2)+(1)/(x^(2)))^(4)

By using properties of determinants, prove that the determinant |(a,sin x,cos x),(-sin x,-a,1),(cos x,1,a)| is independent of x.

int(cos ec^(2)x-2005)/(cos^(2005)x)*dx

Prove that (1- sin 2x)/(1+ sin 2x) = tan^(2) .((pi)/(4)-x)

prove that: sin ^ (4) x + cos ^ (4) x = 1- (1) / (2) sin ^ (2) 2x