JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f: R \rightarrow R\) be a function defined by \(f( x )=( x -3)^{ n _{1}}( x -5)^{ n _{2}}, n _{1}, n _{2} \in N\). The, which of the following is \(\underline{\text { NOT}} \;true? \)
- A For \(n_{1}=3, n_{2}=4\), there exists \(\alpha \in(3,5)\) where \(f\) attains local maxima.
- B For \(n _{1}=4, n _{2}=3\), there exists \(\alpha \in(3,5)\) where \(f\) attains local manima.
- C For \(n _{1}=3, n _{2}=5\), there exists \(\alpha \in(3,5)\) where \(f\) attains local maxima.
- D For \(n_{1}=4, n_{2}=6\), there exists \(\alpha \in(3,5)\) where \(f\) attains local maxima.
Answer & Solution
Correct Answer
(C) For \(n _{1}=3, n _{2}=5\), there exists \(\alpha \in(3,5)\) where \(f\) attains local maxima.
Step-by-step Solution
Detailed explanation
\(f^{\prime}(x)=(x-3)^{n_{1}-1}(x-5)^{n_{2}-1}\left(n_{1}+n_{2}\right)\left(x-\frac{5 n_{1}+3 n_{2}}{n_{1}+n_{2}}\right)\) Option \((3)\) is incorrect since for \(n _{1}=3, n _{2}=5\) \(f^{\prime}(x)=8(x-3)^{2}(x-5)^{4}\left(x-\frac{30}{8}\right)\) \(\operatorname{minima}\) at…
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