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If a twice differentiable function f(x) on (a,b) and continuous on [a, b] is such that f''(x)lt0 for all x in (a,b) then for any c in (a,b),(f(c)-f(a))/(f(b)-f(c))gt |
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Answer» `(B-c)/(c-a)` Then by LMCT on (a, c), (c, b), we have `f'(u)=(f(c)-f(a))/(c-a),f'(v)=(f(b)-f(c))/(b-c)` But `u LT v and f''(x)lt 0, AA x in (a,b)` `"i.e."f'(u)GTF'(v)` `rArr""(f(c)-(a))/(f(b)-f(c))gt(c-a)/(b-c)` |
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