WBJEE · Maths · Application of Derivatives
Let \(\exp (x)\) denote exponential function ex. If \(f(x)=\exp \left(x^{\frac{1}{x}}\right), x>0\) then the minimum value of \(f\) in the interval \([2,5]\) is
- A \(\exp \left(e^{\frac{1}{c}}\right)\)
- B \(\exp \left(2^{\frac{1}{2}}\right)\)
- C \(\exp \left(5^{\frac{1}{5}}\right)\)
- D \(\exp \left(3^{\frac{1}{3}}\right)\)
Answer & Solution
Correct Answer
(C) \(\exp \left(5^{\frac{1}{5}}\right)\)
Step-by-step Solution
Detailed explanation
Given that, \[ f(x)=e^{(x)^{\frac{1}{x}}, x>0} \] Taking log on both sides, we get \[ \log +(x)=(x)^{\frac{1}{x}}=g(x) \] Here, \(g(x)=x^{\frac{1}{x}}\) \(\Rightarrow \quad \log g(x)=\frac{1}{x} \log x\) On differentiating w.r.t. \(x\), we get…
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