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

Options

A. `11//60` sq. units
B. `7//120` sq. units
C. `1//30` sq. units
D. `7//90` sq. units

Explanation

To find the area between the curve \( y = 2x^4 - x^2 \), the x-axis, and the ordinates of the two minima of the curve, we will follow these steps:

### Step 1: Find the critical points of the curve
To find the minima, we first need to determine the critical points by taking the derivative of the function and setting it to zero.

1. Differentiate \( y = 2x^4 - x^2 \):
\[ \frac{dy}{dx} = 8x^3 - 2x \]
2. Set the derivative equal to zero:
\[ 8x^3 - 2x = 0 \]
Factor out \( 2x \):
\[ 2x(4x^2 - 1) = 0 \]
This gives us:
\[ 2x = 0 \quad \text{or} \quad 4x^2 - 1 = 0 \]
Solving \( 4x^2 - 1 = 0 \):
\[ 4x^2 = 1 \quad \Rightarrow \quad x^2 = \frac{1}{4} \quad \Rightarrow \quad x = \pm \frac{1}{2} \]