$2$The use of electric vehicles will curb air pollution in the long run. 2023_CBSE_Mathematics_Set_Region2_Subset_Set2_Q38___p12_1

2023_CBSE_Mathematics_Set_Region2_Subset_Set2_Q38___p12_1

The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time $\displaystyle \mathrm{t}$ is given by the function $\displaystyle \mathrm{V}$ : \[\mathrm{V}(\mathrm{t})=\frac{1}{5} \mathrm{t}^{3}-\frac{5}{2} \mathrm{t}^{2}+25 \mathrm{t}-2 \] where t represents the time and $\displaystyle \mathrm{t}=1,2,3 \ldots$ corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions:

(i)Can the above function be used to estimate number of vehicles in the year 2000 ? Justify.(ii)Prove that the function $\displaystyle \mathrm{V}(\mathrm{t})$ is an increasing function.

Mathematics · Application of Derivatives