Paul uses the conjunction “but” to make a sharp turn in Philippians 4:6. He couldn’t be more clear that anxiety was not an acceptable response for anything – not one thing. So what do we do? The answer is pray. This is the great contrast. Christians disposed to action rejoice that trusting God is not spiritual paralysis. It is not “let go and let God” because the cure for anxiety is never inaction. In fact the Bible teaches the opposite. We are, in everything, to actively engage our minds and our wills. We face off against what scares us by bringing it to the Lord. The most vivid example is Jesus in the garden of gethsemane. This was the most stressful moment of his life, but he is not anxious. Instead he prays. When he got up to prepare his disciples for the advancing mob, he was the picture of calm. He was at peace. His thinking and actions were protected and guided by God and thus remained in perfect step with the will of the Father. This is proof that God does not reward inaction, but will give grace to help in time of need when we pray (Hebrews 4:16).
Update: I just read this blog post from John Piper and it provides a helpful addition to what I wrote above.