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Let f:R rarr R, y=f(x), f(0)=0, f'(x) gt...

Let `f:R rarr R, y=f(x), f(0)=0, f'(x) gt0 and f''(x)gt0`. Three point `A(alpha, f(alpha)), B(beta,f(beta)), C(gamma, f(gamma)) on y=f(x)` such that `0lt alpha lt beta lt gamma.`
Which of the following is false ?

A

`(f(alpha)+f(beta))/(2)lt f((alpha+beta)/(2))`

B

`f(alpha)+f(beta))/(2)gt f((alpha+beta)/(2))`

C

`f(alpha)+f(beta))/(2)=f((alpha+beta)/(2))`

D

`(2f(alpha)+f(beta))/(3)lt f((2alpha+beta)/(3))`

Text Solution

Verified by Experts

The correct Answer is:
B
As the function is concave upward
`(f(alpha)+f(beta))/(2)gtf((alpha+beta)/(2))`
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