WBJEE · Maths · Application of Derivatives
If \(f(x)\) is a function such that \(f^{\prime}(x)=(x-1)^{2}(4-x),\) then
- A \(f(0)=0\)
- B \(f(x)\) is increasing in (0,3)
- C \(x=4\) is a critical point of \(f(x)\)
- D \(f(x)\) is decreasing in (3,5)
Answer & Solution
Correct Answer
(C) \(x=4\) is a critical point of \(f(x)\)
Step-by-step Solution
Detailed explanation
Given \[ f^{\prime}(x)=(x-1)^{2}(4-x) \] The sign scheme of \(f^{\prime}(x)\) Clearly. \(f(x)\) is increasing, for \(x \in(-\infty, 4)\) and decreasing, for \(n \in(4, \infty)\) Since, \(f^{\prime}(x)=0\) at \(x=4\) \(\mathrm{SO}, \mathrm{x}=4\) is a critical point.
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