Question

Value of the parameter 'a' such that the area bounded by y = a²x² + ax + 1, coordinate axes and the line x = 1, attains it's least value, is equal to

• A. −1/4
• B. −1/2
• C. −3/2
• D. -1

Answer 1

Answer: (C)

−3/2

Answer 2

a²x² + ax + 1 is clearly positive for all real values of x. Area under consideration,

$\Delta = \int _ { 0 } ^ { 1 } \left( a ^ { 2 } x ^ { 2 } + a x + 1 \right) d x = \frac { a ^ { 2 } } { 3 } + \frac { a } { 2 } + 1$

$= \frac { 1 } { 6 } \left( 2 a ^ { 2 } + 3 a + 6 \right)$

$= \frac { 1 } { 6 } \left( 2 \left( a ^ { 2 } + \frac { 3 } { 2 } a + \frac { 9 } { 16 } \right) + 6 - \frac { 18 } { 16 } \right)$

$= \frac { 1 } { 6 } \left( 2 \left( a + \frac { 3 } { 4 } \right) ^ { 2 } + \frac { 39 } { 8 } \right)$

Which is clearly minimum for a = −$\frac { 3 } { 4 }$