By R. Haines, B. Haines
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Extra info for Work Out Pure Mathematics A-Level
Sample text
The function Ix I. The solution of inequalities reducible to the form f(x) > 0, where f(x) can then be expressed in factors: sketches of the graphs of y = f(x) in these cases. 1 Fact Sheet (a) Notation: Inequalities • If a and b are numbers, represented by points on the x-axis, and a > b, then a is to the right of b. • Thus 6 > 3 and -6 >--9. b < a 1 • If a ~ b then a is greater than b or equal to b. (b) Modulus • Ix I may be regarded as the distance from the origin to the point on the x-axis .
Thus a < 2 I ~ t for all real x. x + lOx + 27 When x = -5, f(x) = 2, => y = t (maximum point). 4 (a) Show that the arithmetic mean of two positive real numbers is greater than, or equal to, their geometric mean. Hence show that when w, x, y, z are real w4 + X4 + y4 + Z4 ~ 4wxyz. h ' mean • ( a ) The ant metlc y'(wx), where 0 (x E fR,x *0). f w an d . mean i Ii x 'IS w + x ; t h e geometrIC 2 y' means the positive square root. In inequality questions it is often worth considering a bracket squared, for example (w - X)2 .
Ixl<5 => -5
