The string is indeed taut while the particle is above the equilibrium position, since it is still extended beyond its natural length.
To do this correctly by energy considerations (rather than the purely physical argument that I was hinting at), do as follows:
1. At its lowest point, the particle is 2.5 m below the point of attachment. The string has
l=1.5 m. What is the extension of the string? Hence what is the EPE in the string?
Suppose that the particle reaches a point
h above the equilibrium position. At this point, by conservation of energy, some of the EPE that you found in 1. has been converted into PE of the particle. (Draw a picture now to make all of these distances clear in your mind)
2. Write down an expression, in terms of
h, for the gain in PE of the particle from its lowest point.
3. At its highest point, what, in terms of
h, is the extension of the string? Hence write down an expression for the EPE in the string at this point
4. Apply conservation of energy to write down an equation relating the values found in 1), 2), and 3)
5. Solve this equation to find
h