JEE Mains · Maths · STD 12 - 6. Application of derivatives
The position of a moving car at time \(t\) is given by \(f(t)=a t^{2}+b t+c, t>0,\) where \(a, b\) and \(c\) are real numbers greater than \(1 .\) Then the average speed of the car over the time interval \(\left[ t _{1}, t _{2}\right]\) is attained at the point
- A \(a\left(t_{2}-t_{1}\right)+b\)
- B \(\frac{\left( t _{2}- t _{1}\right)}{2}\)
- C \(2 a \left( t _{1}+ t _{2}\right)+ b\)
- D \(\frac{\left( t _{1}+ t _{2}\right)}{2}\)
Answer & Solution
Correct Answer
(D) \(\frac{\left( t _{1}+ t _{2}\right)}{2}\)
Step-by-step Solution
Detailed explanation
\(\frac{f\left(t_{2}\right)-f\left(t_{1}\right)}{t_{2}-t_{1}}=2 a t+b\) \(\frac{a\left(t_{2}^{2}-t_{1}^{2}\right)+b\left(t_{2}-t_{1}\right)}{t_{2}-t_{1}}=2 a t+b\) \(\Rightarrow a\left(t_{2}+t_{1}\right)+b=2 a t+b\) \(\Rightarrow t=\frac{t_{1}+t_{2}}{2}\)
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