TS EAMCET · Maths · Application of Derivatives
Let \(f(x)\) be differentiable on \([1,6]\) and \(f(1)=-2\). If \(f(x)\) has only one root in \((1,6)\) then there exists \(\mathcal{C} \in(1,6)\) such that
- A \(f^{\prime}(c)=\frac{1}{10}\)
- B \(f^{\prime}(\mathrm{c}) < \frac{2}{5}\)
- C \(f^{\prime}(\mathrm{C}) < \frac{1}{5}\)
- D \(f^{\prime}(\mathrm{c})>\frac{2}{5}\)
Answer & Solution
Correct Answer
(D) \(f^{\prime}(\mathrm{c})>\frac{2}{5}\)
Step-by-step Solution
Detailed explanation
\(f(x)\) is differentiable on \([1,6]\) \[ f(1)=-2 \] \(f(x)\) has only are roots in \((1,6)\) \[ \because \] \(f(1) f(6) 0\) By L.M.V.T. \(\quad f^{\prime}(c)=\frac{f(6)-f(1)}{6-1} \Rightarrow f^{\prime}(c)=\frac{f(6)+2}{5}\)…
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