Using Rolle's theorem prove that the equation `3x^2 + px - 1 = 0` has the least one it's interval `(-1,1).`

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

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

If a+b+c=0 , then, the equation 3ax^(2)+2bx+c=0 has , in the interval (0,1).

If 4a+2b+c=0 , then the equation 3ax^(2)+2bx+c=0 has at least one real lying in the interval

Verify Rolle's theorem for the function f(x) = {log (x^(2) +2) - log 3 } in the interval [-1,1] .

The equation x log x = 3-x has, in the interval (1,3) :

Statement-1: If a, b, c in R and 2a + 3b + 6c = 0 , then the equation ax^(2) + bx + c = 0 has at least one real root in (0, 1). Statement-2: If f(x) is a polynomial which assumes both positive and negative values, then it has at least one real root.

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.

Rolle's theorem is applicable for the function f(x) = |x-1| in [0,2] .

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).