TS EAMCET · Maths · Application of Derivatives
Let \(f(x)=e^x \cos x+1\). Which of the following statements is always true?
- A Between any two consecutive roots of \(f(x)=0\) there is always a root of \(e^x \sin x+1=0\)
- B Between any two consecutive roots of \(f(x)=0\) there is always a root of \(e^x \sin x-1=0\)
- C Between any two consecutive roots of \(f(x)=0\) there is always a root of \(e^x \cos x=0\)
- D Between any two consecutive roots of \(f(x)=0\) there is always a roots of \(e^x \sin x=0\)
Answer & Solution
Correct Answer
(D) Between any two consecutive roots of \(f(x)=0\) there is always a roots of \(e^x \sin x=0\)
Step-by-step Solution
Detailed explanation
Let \(f(x)=\cos x+e^{-x}\) and \(\alpha\) and \(\beta\) be two roots of the question \(e^x \cos x+1=0\) such that \(\alpha < \beta\), then \(\cos \alpha+e^{-\alpha}=0\) and \(\cos \beta+e^{-\beta}=0\) clearly, \(f(x)\) is continous on \([\alpha, \beta]\) and differentiable on…
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