If the maximum area of trapezium ABCD formed by points A ( x , 0), B ( x , x +(1/ x )), C (- x , e - x ), D (- x , 0) is S, where 0

Question

If the maximum area of trapezium ABCD formed by points A ( x , 0), B ( x , x +(1/ x )), C (- x , e - x ), D (- x , 0) is S, where 0< x ≤ 1, then find the value of [ eS ]. [Note: [ k ] denote the greatest integer less than or equal to k and ' e ' is napier constant.] (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Area of trapezium Δ=1+ x 2+ xe - x and ( d Δ/ dx ) ≥ 0 when x ∈(0,1] ∴ S =Δ max =2+(1/ e ), when x =1 ∴[ es ]=6 .

Q. If the maximum area of trapezium $A BC D$ formed by points $A (x, 0), B (x, x + \frac{1}{x}), C (- x, e^{- x})$ , $D (- x, 0)$ is $S$ , where $0 < x \leq 1$ , then find the value of $[e S]$ .
[Note : [ $k$ ] denote the greatest integer less than or equal to $k$ and ' $e$ ' is napier constant.]

Area of trapezium $Δ = 1 + x^{2} + x e^{- x}$
and $\frac{d Δ}{d x} \geq 0$ when $x \in (0, 1]$
$∴ S = Δ_{m a x} = 2 + \frac{1}{e}$ , when $x = 1$
$∴ [es] = 6.$

Q. If the maximum area of trapezium ABCD formed by points A(x,0),B(x,x+x1​),C(−x,e−x), D(−x,0) is S, where 0<x≤1, then find the value of [eS]. [Note : [ k ] denote the greatest integer less than or equal to k and ' e ' is napier constant.]

Area of trapezium Δ=1+x2+xe−x and dxdΔ​≥0 when x∈(0,1] ∴S=Δmax​=2+e1​, when x=1 ∴[es]=6.

Q. If the maximum area of trapezium $A BC D$ formed by points $A (x, 0), B (x, x + \frac{1}{x}), C (- x, e^{- x})$ , $D (- x, 0)$ is $S$ , where $0 < x \leq 1$ , then find the value of $[e S]$ .
[Note : [ $k$ ] denote the greatest integer less than or equal to $k$ and ' $e$ ' is napier constant.]

Area of trapezium $Δ = 1 + x^{2} + x e^{- x}$
and $\frac{d Δ}{d x} \geq 0$ when $x \in (0, 1]$
$∴ S = Δ_{m a x} = 2 + \frac{1}{e}$ , when $x = 1$
$∴ [es] = 6.$