For an ideal buck-boost:
V_out = V_in * Dc / ( 1 - Dc)
V_out = 10 * 2/3 / ( 1 - 2/3) = 20 V
Since P_out = P_in and P_out = V_out * I_load:
I_load = P_in / V_out
I_load = 100 / 20 = 5 A
We can devide one period by 3; equal intervals
In 2 intervals, the capacitor is discharged by the load with a current 5 A
In 1 interval, the capacitor should be charged with a current so that:
I_chg * t_chg = I_dis * t_dis
or
I_chg = I_dis * t_dis / t_chg
But the ratio t_dis / t_chg = 2 / 1 = 2
I_chg = 5 * 2 = 10 A
So for 2 intervals the current is Ic1 = 5 A
And for 1 interval the current is Ic2 = 10 A
Their signs (positive and negative) are not important here since we will take their square.
Ic_rms = SQRT { [ (Ic1^2)*2 + (Ic2^2) * 1 ] / 3 }
Ic_rms = SQRT { [ (5^2)*2 + (10^2)*1 ] / 3 }
Ic_rms = SQRT { [ 50 + 100 ] / 3 } = SQRT(50)
Ic_rms = 7.07 A
I hope it is clear.