If `f(x) = xlnx - 2,` then show that `f(x) = 0` has exactly one root in the interval `(1, e).`

If f(x)=1+x^(m)(x-1)^(n), m , n in N , then f'(x)=0 has atleast one root in the interval

For which values of 'a' so that the equation x^(2)+(3-2a)x+a=0 has exactly one root in the interval (-1,2)

Consider the polynomial f(x)=1 + 2x + 3x^2 +4x^3 for all x in R So f(x) has exactly one real root in the interval

Let f(x) =x^(2) , x in (-1,2) Then show that f(x) has exactly one point of local minima but global maximum Is not defined.

Let f (x) = x^(2) , x in [-1 , 2) Then show that f(x) has exactly one point of local minima but global maximum is not defined.

If f'(x)=0 for all values of x in an interval, then show that f(x) is constant in that interval.

Find the value of l'al'so that the equation f(x)=x^(2)+(a-3)x+a=0 has exactly one alpha between the interval (1,2) and f(x+alpha)=0 has exactly one root between the interval (0,1)

Find the value of l'al'so that the equation f(x)=x^(2)+(a-3)x+a=0 has exactly one alpha between the interval (1,2) and f(x+alpha)=0 has exactly one root between the interval (0,1)

Prove that the equation x^3 + 2x^2 +x + 4 = 0 has exactly one root in the open interval (-3,-2).

Without actually solving show that the equation x^(4)+2x^(3)-2=0 has only one real root in the interval (0,1) .