Let’s use LMVT for x∈[a,c]
c−af(c)−f(a)=f′(α),α∈(a,c)
Also use LMVT for x∈[c,b]
b−cf(b)−f(c)=f′(β),β∈(c,b)
∵f′′(x)<0⇒f′(x) is decreasing
f′(α)>f′(β)
c−af(c)−f(a)>b−cf(b)−f(c)
f(b)−f(c)f(c)−f(a)>b−cc−a ( ∵f(x) is increasing)
Let f be any function continuous on [a,b] and twice differentiable on (a,b) . If all x∈(a,b),f′(x)>0 and f′′(x)<0 , then for any c∈(a,b),f(b)−f(c)f(c)−f(a)
Held on 9 Jan 2020 · Verified 6 Jul 2026.
b−ab+a
1
c−ab−c
b−cc−a
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