The sentence that requires semicolons instead of commas is:
A well-balanced diet includes root vegetables, such as carrots, dark red fruits, such as strawberries, and whole grains, such as brown rice.
In this sentence, the ideas presented after each phrase could get confusing due to the multiple uses of commas. Each "such as" phrase can be considered a separate item in a complex list, so semicolons can help clarify the separation of these items.
The corrected sentence with semicolons should read:
A well-balanced diet includes root vegetables, such as carrots; dark red fruits, such as strawberries; and whole grains, such as brown rice.