Application Of Integration

Suppose we define the definite integral using the following formula \(\int_\limits{a}^{b} f(x) d x=\frac{b-a}{2}(f(a)+f(b))\), for more accurate result for

\(c \in(a, b) \mathrm{F}(c)=\frac{c-a}{2}(f(a)+f(c))+\frac{b-c}{2}(f(b)+f(c))\).

When \(c=\frac{a+b}{c}, \int_\limits{a}^{b} f(x) d x=\frac{b-a}{4}(f(a)+f(b)+2 f(c))\)

\(f''(x) < 0 \forall x \in(a, b)\) and \(c\) is a point such that \(a < c < b\), and \((c, f(C))\) is the point lying on the curve for which \(\mathrm{F}(C)\) is maximum, then \(f'(C)\) is equal to: