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A wire fixed at the upper end stretches by length $$\ell$$ by applying a force $$F$$. The work done in stretching is
Since the force increases linearly from $$0$$ to $$F$$, the average force applied to the wire is:
$$F_{avg} = \frac{0 + F}{2} = \frac{F}{2}$$
The total work done ($$W$$) is the product of this average force and the total extension ($$\ell$$):
$$W = F_{avg} \times \text{extension}$$
$$W = \left(\frac{F}{2}\right) \times \ell$$
$$W = \frac{1}{2} F \ell$$
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