Let \(f\) be a polynomial function defined on \([2,7]\). If \(f(2)=3\) and \(f^{\prime}(x) \leq 5\) for all…
Let \(f\) be a polynomial function defined on \([2,7]\). If \(f(2)=3\) and \(f^{\prime}(x) \leq 5\) for all \(x\) in \((2,7)\), then the maximum possible value attained by \(f\) at \(x=7\) is
7
14
18
28
Solution
Since, the polynomial function are continuous and differentiable in interval \(R\).
So, according to Lagranage's mean value theorem
\(\begin{aligned}
& \frac{f(7)-f(2)}{7-2}=f^{\prime}(x) \leq 5 \quad \text{(given)} \\
& \Rightarrow \quad f(7)-f(2) \leq 25 \Rightarrow f(7) \leq 28
\end{aligned}\)
Hence, option (d) is correct.