Class 12 Continuity And Differentiability · Case Study4 Markscase_based_application
Class 12 Maths · Continuity And Differentiability · Case Study #35
A roller-coaster path is modeled by the polynomial f(x)=a(x+9)(x+1)(x−3). It crosses the y-axis at (0,−1). Use the mathematical model, rather than the indicative physical appearance, for the calculations.
Part (1) 2 Marks
Find a.
Answer: 1/27
- At x=0, f(0)=a·9·1·(−3)=−27a.
- Since f(0)=−1, −27a=−1 and a=1/27.
- Therefore a=1/27.
Part (2) 2 Marks
Find f″(1).
Answer: 20/27
- Expand the model using a=1/27: f(x)=(x³+7x²−21x−27)/27.
- Differentiate twice: f′(x)=(3x²+14x−21)/27 and f″(x)=(6x+14)/27.
- Thus f″(1)=(6+14)/27=20/27.