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if $\displaystyle \int^{x}_{a}\Bigg(\bigg(\frac{f(t)-f(a)}{(t-a)^2}\bigg)-\frac{f'(-a)}{(t-a)}\Bigg)dt$ has no critical points, then $f(x)$ can be

$(a)\; x+\sin x$

$(b)\; x^3+3x+2$

$(c)\; x^2+\sin x$

$(d)\; x^2-\sin x$

Attempt if $\displaystyle g(x)=\int^{x}_{a}\Bigg(\bigg(\frac{f(t)-f(a)}{(t-a)^2}\bigg)-\frac{f'(-a)}{(t-a)}\Bigg)dt$ has no critical point

then $g(x)$ is strictly increasing function or strictly decreasing function

so $\displaystyle g'(x) = \frac{f(x)-f(a)}{(x-a)^2}-\frac{f'(-a)}{x-a}>0$ or $<0$

wan,t be able to go further could some helpme, thanks

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    Is $f$ the integral? – 2017-01-16
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    Look at the denominators in the integrand $\frac{f(t)-f(a)}{(t-a)^2}-\frac{f'(-a)}{(t-a)}$ and then look at your expression for $g'(x)$. I think there's been a faulty cancelling somewhere there. – 2017-01-16

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