A) 0
B) \[\frac{1}{4}\]
C) \[\frac{1}{2}\]
D) \[\frac{1}{3}\]
Correct Answer: B
Solution :
Let \[f(x)={{x}^{25}}{{(1-x)}^{75}}\] On differentiating w. r. t. x, we get \[f'(x)=25{{x}^{24}}{{(1-x)}^{75}}-75{{x}^{25}}{{(1-x)}^{74}}\] \[=25{{x}^{24}}[{{(1-x)}^{74}}(1-x-3x)]\] \[=25{{x}^{24}}{{(1-x)}^{74}}(1-4x)\] \[\therefore \] \[f'(x)\] changes sign about \[x=\frac{1}{4}\] only.
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