PureMath.pdf

# 3 show that x sin x is an increasing function

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3. Show that x - sin x is an increasing function throughout any interval of values of x , and that tan x - x increases as x increases from - 1 2 π to 1 2 π . For what values of a is ax - sin x a steadily increasing or decreasing function of x ? 4. Show that tan x - x also increases from x = 1 2 π to x = 3 2 π , from x = 3 2 π to x = 5 2 π , and so on, and deduce that there is one and only one root of the equation tan x = x in each of these intervals (cf. Ex. xvii . 4). 5. Deduce from Ex. 3 that sin x - x < 0 if x > 0, from this that cos x - 1 + 1 2 x 2 > 0, and from this that sin x - x + 1 6 x 3 > 0. And, generally,

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[VI : 124] DERIVATIVES AND INTEGRALS 269 prove that if C 2 m = cos x - 1 + x 2 2! - · · · - ( - 1) m x 2 m (2 m )! , S 2 m +1 = sin x - x + x 3 3! - · · · - ( - 1) m x 2 m +1 (2 m + 1)! , and x > 0, then C 2 m and S 2 m +1 are positive or negative according as m is odd or even. 6. If f ( x ) and f 00 ( x ) are continuous and have the same sign at every point of an interval [ a, b ], then this interval can include at most one root of either of the equations f ( x ) = 0, f 0 ( x ) = 0. 7. The functions u , v and their derivatives u 0 , v 0 are continuous throughout a certain interval of values of x , and uv 0 - u 0 v never vanishes at any point of the interval. Show that between any two roots of u = 0 lies one of v = 0, and conversely. Verify the theorem when u = cos x , v = sin x . [If v does not vanish between two roots of u = 0, say α and β , then the function u/v is continuous throughout the interval [ α, β ] and vanishes at its extremities. Hence ( u/v ) 0 = ( u 0 v - uv 0 ) /v 2 must vanish between α and β , which contradicts our hypothesis.] 8. Determine the maxima and minima (if any) of ( x - 1) 2 ( x + 2), x 3 - 3 x , 2 x 3 - 3 x 2 - 36 x + 10, 4 x 3 - 18 x 2 + 27 x - 7, 3 x 4 - 4 x 3 + 1, x 5 - 15 x 3 + 3. In each case sketch the form of the graph of the function. [Consider the last function, for example. Here φ 0 ( x ) = 5 x 2 ( x 2 - 9), which vanishes for x = - 3, x = 0, and x = 3. It is easy to see that x = - 3 gives a maximum and x = 3 a minimum, while x = 0 gives neither, as φ 0 ( x ) is negative on both sides of x = 0.] 9. Discuss the maxima and minima of the function ( x - a ) m ( x - b ) n , where m and n are any positive integers, considering the different cases which occur according as m and n are odd or even. Sketch the graph of the function. 10. Discuss similarly the function ( x - a )( x - b ) 2 ( x - c ) 3 , distinguishing the different forms of the graph which correspond to different hypotheses as to the relative magnitudes of a , b , c . 11. Show that ( ax + b ) / ( cx + d ) has no maxima or minima, whatever values a , b , c , d may have. Draw a graph of the function.
[VI : 124] DERIVATIVES AND INTEGRALS 270 12. Discuss the maxima and minima of the function y = ( ax 2 + 2 bx + c ) / ( Ax 2 + 2 Bx + C ) , when the denominator has complex roots. [We may suppose a and A positive. The derivative vanishes if ( ax + b )( Bx + C ) - ( Ax + B )( bx + c ) = 0 . (1) This equation must have real roots. For if not the derivative would always have the same sign, and this is impossible, since y is continuous for all values of x , and y a/A as x + or x → -∞ . It is easy to verify that the curve cuts the line y = a/A in one and only one point, and that it lies above this line for large positive values of x , and below it for large negative values, or

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