Let `y =f (x)` be a twice differentiable, non- negative function defined on `[a.b].` The area `int _(a) ^(b) f(x) dx, b gt` a bounded by `y = f(x),` the x-axis and the ordinates at `x = a and x = b` can be approximated as
`int _(a ) ^(b) f (x) dx ~~((b-a))/( 2) {f(a) + f (b)}.`
Since `int _(a)^(b) f (x) dx = int _(a) ^(c ) f (x) dx + int _(c ) ^(b) f (x) dx, x hatI (a,b) , a ` better approximation to `int _(a) ^(b) f(x) dx` can be written as `int _(a) ^(b)f (x) dx "@" ((c-a))/(2) {f(a)+f(c )}_ ((b-c))/(2)`
`{f(c )+ f (b)} ""^(@)F(c ),`
If `c = (b-a)/(2),` then this gives:
`int _(a) ^(b )f (x) dx "@" (b-a)/(4) {f(a) + 2f (a) + 2f (c )+ f(b) },.......(1)`
If `f ''(x) lt 0, x in (a,b),` then at the point
`C(c, f (c))on y = f (x) ` for which F(c ) is a maximum,
`f '(c ) ` is given by

Options

A. `f'(c ) (f (b) -f (a))/(b-a)`
B. `f ' (c ) (f (b) - f(a))/(a-b) `
C. ` f ' (c ) = (2 (f (b) -f (a)))/(b-a)`
D. `f ' (c ) =0`

Answer

A