The area between the curve y = 2 x 4 - x 2 , the x -axis and the ordinates of the two minima of the curve is

Step-by-step solution

The equation of the curve is y = 2 x 4 - x 2 = 2 x 2 - 1 x 2 The curve is symmetrical about the y -axis. Also, it is a polynomial of degree four having roots 0 , 0 , ± 1 2 . x = 0 is repeated root. Hence, graph touches x -axis at (0,0) and intersects the x -axis at A − 1 2 , 0 & B 1 2 , 0 ⇒ dy dx = 8 x 3 − 2 x = 2 x ( 4 x 2 − 1 ) = 0 x = 0 , x = ± 1 2 d 2 y d x 2 > 0 at x = 1 2 and at x = − 1 2 So x = ± 1 2 are the points of local minima Thus, the graph of the curve is shown in diagram. Here, y ≤ 0 , as x varies fr