What is the justification for relying on faith healing and conventional medicine? As a contrast to What is the justification for relying on faith healing over conventional medicine?, I thought I'd ask the opposite. What is the justification for relying on faith and conventional medicine in conjunction with one another?
In Like a Broken Vessel, Elder Jeffrey R. Holland is speaking to members of the LDS Church about mental illness, but I think his reasoning is more generally applicable.


  If you had appendicitis, God would expect you to seek a priesthood
  blessing and get the best medical care available. ...
  Our Father in Heaven expects us to use all of the
  marvelous gifts He has provided in this glorious dispensation.


So Elder Holland is making the argument that since God has provided us with both faith (and in the case of latter-day saints, blessings of healing) and with medical care, we should use both.

President Dallin H. Oaks goes into more detail in Healing the Sick


  The use of medical science is not at odds with our prayers of faith
  and our reliance on priesthood blessings. When a person requested a
  priesthood blessing, Brigham Young would ask, “Have you used any
  remedies?” To those who said no because “we wish the Elders to lay
  hands upon us, and we have faith that we shall be healed,” President
  Young replied: “That is very inconsistent according to my faith. If we
  are sick, and ask the Lord to heal us, and to do all for us that is
  necessary to be done, according to my understanding of the Gospel of
  salvation, I might as well ask the Lord to cause my wheat and corn to
  grow, without my plowing the ground and casting in the seed. It
  appears consistent to me to apply every remedy that comes within the
  range of my knowledge, and [then] to ask my Father in Heaven … to
  sanctify that application to the healing of my body.”
  
  Of course we don’t wait until all other methods are exhausted before
  we pray in faith or give priesthood blessings for healing. In
  emergencies, prayers and blessings come first. Most often we pursue
  all efforts simultaneously. This follows the scriptural teachings that
  we should “pray always” (D&amp;C 90:24) and that all things should be done
  in wisdom and order.


President Oaks is making it clear that seeking medical attention (if possible) is a sign of faith, and therefore a prerequisite for a miracle to occur. This is especially important in our day, where many of us have access to excellent medical care. That being said, even in biblical times they did the best they could. Ancient Israel encouraged a healthy diet. In 1 Corinthians 3:16-17, it states that we are the temple of God, which most not be defiled.

I think that many Christians who rely on faith and conventional medicine would agree for broadly similar reasons.