WBJEE · Maths · Application of Derivatives
If \(f(x)=e^{x}(x-2)^{2},\) then
- A \(f\) is increasing in \((-\infty, 0)\) and \((2, \infty)\) and decreasing \(\operatorname{lin}(0,2)\)
- B \(t\) is increasing in \((-\infty, 0)\) and decreasing in \((0, \infty)\)
- C \(f\) is increasing in \((2, \infty)\) and decreasing in \((-\infty, 0)\)
- D \(f\) is increasing in (0,2) and decreasing in \((-\infty, 0)\) and \((2, \infty)\)
Answer & Solution
Correct Answer
(A) \(f\) is increasing in \((-\infty, 0)\) and \((2, \infty)\) and decreasing \(\operatorname{lin}(0,2)\)
Step-by-step Solution
Detailed explanation
Given function is, \(f(x)=e^{x}(x-2)^{2}\) \[ \begin{aligned} \Rightarrow f^{\prime}(x) &=e^{x}(x-2)^{2}+2(x-2) e^{x} \\ &=e^{x}(x-2)(x-2+2)=x(x-2) e^{x} \end{aligned} \] Now, sign scheme of \(f^{\prime}(x)\) is So, \(t\) is increasing in \((-\infty, 0)\) and \((2, \infty)\) and…
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