One relationship between marginal utility and total utility is that when total utility is

Marginal utility (MU) is the additional satisfaction a consumer gains from consuming one more unit of a good. Total utility (TU) is the cumulative satisfaction from all units consumed. The relationship between them follows a predictable pattern governed by the law of diminishing marginal utility:

• When MU is positive (but declining), each additional unit still adds to total satisfaction, so TU is rising.
• When MU reaches zero, the consumer gains no extra satisfaction from the next unit, and TU is at its maximum.
• When MU becomes negative, additional consumption actually reduces satisfaction, and TU begins to fall.

Therefore, when total utility is falling, marginal utility is negative. This is the correct relationship.

The statement that when TU is maximum, MU is maximum is incorrect - at maximum TU, MU equals zero, not maximum. The claim that when TU is rising, MU is also rising is also incorrect - TU can rise while MU is falling (diminishing), which is exactly what the law of diminishing marginal utility describes. The statement that when TU is falling, MU is rising is contradictory and impossible under the standard utility framework.