Question

The area bounded by the curve y = x⁴ – 2x³ + x² + 3, the axis of abscissas and two ordinates corresponding to the points of minimum of the function y (x) is –

• A. 10/3
• B. 27/10
• C. 21/10
• D. None of these

Answer 1

Answer: (B)

27/10

Answer 2

Given curve is y = x⁴ – 2x³ + x² + 3 \ $\frac { d y } { d x }$ = 4x³ – 6x² + 2x = 12x² – 12x + 2 For maxima and minima = 0 \ 4x³ – 6x² + 2x = 0 then x = 0, $\frac { 1 } { 2 }$ , 1

\ = 2, = – 1 nd = 2

\ Points of minimum are x = 0 and x = 1

\ Required area = $\int _ { 0 } ^ { 1 } \left( x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } + 3 \right) d x$

= $\left[ \frac { x ^ { 5 } } { 5 } - \frac { 2 x ^ { 4 } } { 4 } + \frac { x ^ { 3 } } { 3 } + 3 x \right] _ { 0 } ^ { 1 }$

= $\frac { 1 } { 5 } - \frac { 2 } { 4 } + \frac { 1 } { 3 } + 3$ = $\frac { 1 } { 5 } - \frac { 1 } { 2 } + \frac { 1 } { 3 } + 3$= $\frac { 27 } { 10 }$sq. units.