`(x+1)(x+2) = (x+2)(x+3) + (x+3)(x+4)`

Simplyfy: (3x^(2) + 5x - 7)(x -1)- (x^(2) - 2x +3)(x + 4)

If f'(x)=(x-1)(x-2)^(2)(x-3)^(3)(x-4)^(4)(x-5)^(5) , then the number of local minima of f(x) is

If f'(x)=(x-1)(x-2)^(2)(x-3)^(3)(x-4)^(4)(x-5)^(5) , then the number of local minima of f(x) is

Check whether the following are quadratic equation (i) (x+1)^2=2(x-3) (ii) x^2-2x=(-2)(3-x) (iii) (x-2)(x+1)=(x-1)(x+3) (iv) (x-3)(2x+1)=x(x+5) (v) (2x-1)(x-3)=(x+5)(x-1) (vi) x^2+3x+1=(x-2)^2 (vii) (x+2)^3=2x(x^2-1) (viii) x^3-– 4x^2 – x + 1 = (x – 2)^3

The equation x ^(4) -2x ^(3)-3x^2 + 4x -1=0 has four distinct real roots x _(1), x _(2), x _(3), x_(4) such that x _(1) lt x _(2) lt x _(3)lt x _(4) and product of two roots is unity, then : x _(1)x _(2) +x_(1)x_(3) + x_(2) x _(4) +x_(3) x _(4)=

Solvef or x,(1)/((x-1)(x-2))+(1)/((x-2)(x-3))+(1)/((x-3)(x-4))=(1)/(6)