Let f: R → R be a function defined by f(x) = (x - 3) ^ n _ 1 (x -...

Given,

$f\left(x\right) = {\left(x - 3\right)}^{{n}_{1}}{\left(x - 5\right)}^{{n}_{2}}$

Differentiating the function to get maxima and minima, we get

$f ' \left(x\right) = {\left(x - 3\right)}^{{n}_{1} - 1}{\left(x - 5\right)}^{{n}_{2} - 1}\left({n}_{1} + {n}_{2}\right)\left(x - \dfrac{5{n}_{1} + 3{n}_{2}}{{n}_{1} + {n}_{2}}\right)$

Now checking the option, we get

Option (3) is incorrect since

for ${n}_{1} = 3, {n}_{2} = 5$

$f ' \left(x\right) = 8{\left(x - 3\right)}^{2}\left(x - 5\right){}^{4}\left(x - \dfrac{30}{8}\right)$

minima at $x = \dfrac{30}{8}$.