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Q. Suppose that $\mathrm{f}(0)=-3$ and $\mathrm{f}^{\prime}(\mathrm{x}) \leq 5$ for all values of $\mathrm{x}$. Then the largest value which $\mathrm{f}(2)$ can attain is

Application of Derivatives

Solution:

Using LMVT in $[0,2]$
$ \frac{\mathrm{f}(2)-\mathrm{f}(0)}{2-0}=\mathrm{f}^{\prime}(\mathrm{c}) \text { where } \mathrm{c} \in(0,2)$
$ \frac{\mathrm{f}(2)+3}{2} \leq 5$
$f (2) \leq 7$