Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
If =7 x^{2} e^{-x^{2}} \forall x \in R, \) then \) has This question has multipl
If \( g(x)=7 x^{2} e^{-x^{2}} \forall x \in R, \) then \( g(x) \) has This question has multiple correct options
- A. local maximum at \( x=0 \)
- B. local minima at \( x=0 \)
- C. local maximum at \( x=1 \)
- D. two local maxima and one local minima
Step-by-step solution
Derivative g'(x)=14x e^{-x^2}(1-x^2). Critical points at x=0, ±1. Sign analysis shows g' changes from + to - at x=-1 (max), - to + at x=0 (min), + to - at x=1 (max). Hence g has two local maxima at x=±1 and one local minimum at x=0.
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