A curve is given between potential energy of a particle and its position on x-axis.

Given: tanθ₁ = 1, tanθ₂ = 3, tanθ₃ = −1/2

If F_AB be force acting on the particle during A → B similarly F_BC, F_CD and F_DE are the forces during B → C, C → D and D → E respectively. Arrange magnitudes of these forces in decreasing order.

The correct option is F_BC > F_AB > F_DE > F_CD.

1. Understanding the Relationship Between Force and Potential Energy:
The conservative force $F$ acting on a particle is related to its potential energy $U$ as a function of position $x$ by the following formula:
$F=-\frac{dU}{dx}$
Here, $\frac{dU}{dx}$ represents the slope of the potential energy curve $U$ versus position $x$.

Therefore, the magnitude of the force is equal to the absolute value of the slope of the curve in each region:
$\left|F\right|=\left|\frac{dU}{dx}\right|=\left|\text{slope}\right|$

2. Analyzing Each Segment of the Curve:
Looking at the provided graph, we can determine the magnitude of the force in each region based on the given trigonometric tangent values of the angles that these line segments make with the horizontal:

• Region A → B: The line segment AB makes an angle ${\theta }_1$ with the horizontal. The slope of this segment is:
  ${\text{Slope}}_{\text{AB}}=tan{\theta }_1=1$
  Thus, the magnitude of the force is:
  $\left|F_{\text{AB}}\right|=\left|1\right|=1$
  
• Region B → C: The line segment BC makes an angle ${\theta }_2$ with the horizontal. The slope of this segment is:
  ${\text{Slope}}_{\text{BC}}=tan{\theta }_2=3$
  Thus, the magnitude of the force is:
  $\left|F_{\text{BC}}\right|=\left|3\right|=3$
  
• Region C → D: The line segment CD is horizontal (parallel to the x-axis), which means its slope is zero:
  ${\text{Slope}}_{\text{CD}}=0$
  Thus, the magnitude of the force is:
  $\left|F_{\text{CD}}\right|=0$
  
• Region D → E: The line segment DE makes an angle ${\theta }_3$ with the positive x-axis. The slope of this segment is:
  ${\text{Slope}}_{\text{DE}}=tan{\theta }_3=-\frac{1}{2}$
  Thus, the magnitude of the force is:
  $\left|F_{\text{DE}}\right|=|-\frac{1}{2}|=0.5$
  

3. Arranging in Decreasing Order:
Comparing the calculated force magnitudes:
$\left|F_{\text{BC}}\right|=3$
$\left|F_{\text{AB}}\right|=1$
$\left|F_{\text{DE}}\right|=0.5$
$\left|F_{\text{CD}}\right|=0$
Since $3>1>0.5>0$, the decreasing order of the magnitudes of the forces is:
$F_{\text{BC}}>F_{\text{AB}}>F_{\text{DE}}>F_{\text{CD}}$