Show that
(a) `x^(2)-3x+6 gt 0` for all x,
(b) `4x-x^(2)-6 lt 0` for all x.
(c) `2x^(2)-4x+7` is always +ve.
(d) `-2x^(2)+3x-4` is always -ve.
(e) `-x^(2)+3x-3` is always -ve.

If x is real, prove that the quadratic expression (i) (x-2)(x+3)+7 is always positive. (ii) 4x-3x^(2)-2 is always negative.

If x^(2)+6x-27 gt 0 and x^(2)-3x-4 lt 0 , then :

If f'(x^2-4x+3)gt0 " for all " x in (2,3) then f(sinx) is increasing on

If x is real show sthat the value of (x^(2)+2x+3)/(x^(2)+2x+4) always lies between 2/3 and 1.

Product of real roots of the equation t^2x^2+|x|+9=0 a. is always +ve b. is always-ve c. does not exist d. none of these

Product of real roots of the equation t^2x^2+|x|+9=0 a. is always +ve b. is always-ve c. does not exist d. none of these

Find the roots of the following quadratic equations i) 6sqrt5 x^(2) – 9x -3sqrt5 = 0 ii) x^(2) - x - 12 = 0 iii) 2x^(2) - 6x + 7 = 0 iv) 4x^(2) - 4x+17 = 3x^(2) -10x-17 v) x^(2) + 6x + 34 = 0 vi) 3x^(2) + 2x - 5 = 0

The set of all x satisfying 3^(2x)-3x^(x)-6gt0 is given by

Show that 2x + 7 is a factor of 2x^3 + 7x^2 - 4x - 14 Hence, solve the equation: 2x^3 + 7x^2 - 4x - 14 = 0.

The function f(x) ={{:( 5x-4, " for " 0 lt x le 1) ,( 4x^(2) - 3x, " for" 1 lt x lt 2 ),( 3x + 4 , "for" x ge2):}is