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if f'(x) is continuous in [a,b] and differentiable in (a,b). Show that there exist a c in (a,b) such that f(b)=f(a)+(b-a)f'(a)+(b-a)^2f''(c)/2

if f'(x) is continuous in [a,b] and differentiable in (a,b). Show that there exist a c in (a,b) such that f(b)=f(a)+(b-a)f'(a)+(b-a)^2f''(c)/2

Grade:12

1 Answers

Aditya Gupta
2081 Points
3 years ago
consider g(x)= f(b) – f(x) – f’(x)(b – x). obviously g’(x)= (x – b)f”(x)
now define h(x) = g(x) – [(b – x)/(b – a)]^2*g(a)
note that h(a)= h(b)= 0
so apply rolles theorem on h.
h’(c)=0 for some c in (a, b)
or g’(c)= 2(c – b)g(a)/(b – a)^2
or (c – b)f”(c)= 2(c – b)g(a)/(b – a)^2
or (b – a)^2f”(c)/2= g(a)
or (b – a)^2f”(c)/2=  f(b) – f(a) – f’(a)(b – a)
or f(b)=f(a)+(b-a)f'(a)+(b-a)^2f''(c)/2
kindly approve :)

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