The set of all points for which f(x) = x^2 cd | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the function $f(x) = x^2 e^{-x}$.To find the interval where the function is strictly increasing,we find its derivative $f'(x)$:$f'(x) = \fra

Vedclass · Accepted Answer

$(0, 2)$

Vedclass · Answer

(A) Given the function $f(x) = x^2 e^{-x}$.To find the interval where the function is strictly increasing,we find its derivative $f'(x)$:$f'(x) = \frac{d}{dx}(x^2) \cdot e^{-x} + x^2 \cdot \frac{d}{dx}(e^{-x})$$f'(x) = 2x e^{-x} - x^2 e^{-x}$$f'(x) = x e^{-x} (2 - x)$For the function to be strictly increasing,we must have $f'(x) > 0$.Since $e^{-x}$ is always positive for all real $x$,the sign of $f'(x)$ depends on $x(2 - x)$.$x(2 - x) > 0$Multiplying by $-1$ reverses the inequality:$x(x - 2) The roots are $x = 0$ and $x = 2$. Testing the intervals $(-\infty, 0)$,$(0, 2)$,and $(2, \infty)$:For $x \in (0, 2)$,$x$ is positive and $(x - 2)$ is negative,so $x(x - 2)  0$.Thus,the function $f(x)$ is strictly increasing in the interval $(0, 2)$.

The set of all points for which $f(x) = x^2 \cdot e^{-x}$ strictly increases is

Similar Questions

In Young's double slit experiment,one of the slits is wider than the other,so that the amplitude of the light from one slit is double that from the other slit. If $I_m$ be the maximum intensity,the resultant intensity when they interfere at phase difference $\phi$ is given by:

Which one of the following statements is correct with respect to basic character?

Crystalline sodium chloride is an insulator of electricity because $......$

The major product in the following reaction is:

The $Dystrophin$ gene contains how many million bases?

Vedclass Test Series

Exam Paper Generator

Online Exam Module