Two formulas can meet without making a smooth join.
WHAT HAPPENS IF…Is b=0 enough to make the join differentiable?
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Connect the picture to the ideaf(x)={x2mx+bx<0x≥0
The left-hand limit at zero is 0 and the right-hand value and limit are b. Therefore continuity requires b=0. The left derivative is 0 and the right derivative is m, so differentiability additionally requires m=0.