CHAPTER VII. 
ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND 
, INTEGRAL CALCULUS. 
129. Higher Mean Value Theorems. In the preceding 
chapter we proved (§ 106) that if f(x) has a derivative f'(x) 
throughout the interval (a, b), then 
f{h) -j (a) = (b - a) f {a + 0 {h - a)} 
where 0 lies between 0 and 1; or, if b = a + h, 
f{a + h)-J(a) = hf'(a + 0h) (1). 
This we proved by considering the function 
f(b) -/(«)-1/(6) -/(«)) 
which vanishes for x = a and x = b. 
Let us now suppose that f(x) has also a second derivative 
/» throughout (a, b) (an assumption which of course involves 
the continuity of the first derivative f'(x)), and consider the 
function 
f(b)-f{x) -{b- x)f(x) - i/( 6 > “ /(°) ~ ( b ~ «)/'(«))• 
This function also vanishes for x = a and x—b\ and its 
derivative is 
~/( ft ) - ( h ~ «)/'(«) “ 2 i b ~ aff\x)], 
and this must vanish (§ 102) for some value of x between a and b 
(exclusive of a and b). Hence there is a value £ of x, between 
a and b, and therefore capable of representation in the form 
a + 0 (b — a), where 0 < 0 < 1, which is such that 
f{b) =f(a) + {b- a)/'(a) + № ~ «) 2 /" (£)•