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Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

The normals to the curve 1, drawn at the points with the abscissa and

The normals to the curve \( y=x^{2}-x+ \) 1, drawn at the points with the abscissa \( \boldsymbol{x}_{1}=\mathbf{0}, \boldsymbol{x}_{2}=-\mathbf{1} \) and \( \boldsymbol{x}_{3}=\frac{\mathbf{5}}{\mathbf{2}} \)

  • A. are parallel to each other
  • B. are pair wise perpendicular
  • C. are concurrent
  • D. are not concurrent

Step-by-step solution

For y = x^2 - x + 1, dy/dx = 2x - 1. Slopes of normals: at x1=0: m1 = -1/(-1)=1; at x2=-1: m2 = -1/(-3)=1/3; at x3=5/2: m3 = -1/4. Points: P1(0,1), P2(-1,3), P3(5/2,19/4). Equations: Normal1: y = x + 1; Normal2: y = (1/3)x + 10/3; Normal3: y = (-1/4)x + 43/8. Solving Normal1 and Normal2 gives intersection (3.5, 4.5). Substituting into Normal3 gives 4.5 = (-1/4)*3.5 + 43/8 = 4.5, so all three normals pass through (3.5, 4.5). Thus they are concurrent.
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