Concept:
For a function y = f(x):
- At the relative (local) extrema (maxima or minima), f'(x) = 0.
- In the regions where f(x) is increasing, f'(x) > 0.
- In the regions where f(x) is decreasing, f'(x) < 0.
Formula:
- dxdxn=nxn−1.
- dxdex=ex.
- dxd(uv)=udxdv+vdxdu.
Calculation
Let us first find dxdy for the given function y=2x3ex.
dxd(2x3ex)=2x3(dxdex)+2ex(dxdx3)
=2x3ex+6x2ex
=2x2ex(x+3)
⇒2x2ex(x+3)=0
Since, e^{\mathbf{x}}>0 for all values of x, we get:
⇒x2=0ORx+3=0
⇒x=0 OR x=-3.
So, we need to analyze the function in the intervals (->,-3),(-3,0) and (0, >).
In the interval (0,∞):
\overline{{\mathbf{f}(1)=2\mathbf{x}^{2}\mathbf{e}^{\mathbf{x}}(\mathbf{x}+3)=8\mathbf{e}>0.}}
∴f(x) is increasing in (0,∞).
In the interval (-3,0):
\mathrm{f(-1)=2(-1)^{2}e^{-1}(-1+3)=\frac{4}{e}>0.}
∴f(x) is increasing in (-3,0)
In the interval (-∞,−3)
\overline{{\mathrm{f}(-4)=2(-4)^{2}\mathrm{e}^{-4}(-4+3)=-\frac{32}{\mathrm{e}^{4}}<0.}}
∴f(x) is decreasing in(−∞,−3)
We can see that the function is increasing in the intervals (-3,0) and (0,x).
