Knowing the function $f(x)=e xsin(x)x 32x$, find the extreme point of this function in the definition domain $mathbb$.
Solution: Step 1: Determine the defined domain.
In the function $f(x)=e xsin(x)x 32x$, the domains of $e x$ and $sin(x)$ are all real numbers $mathbb$, so the domain of this function is also $mathbb$.
Step 2: Find the derivative.
Derivative of the function $f(x)$ yields:
f'(x)=e^x(sin(x)cos(x))3x^22$$。
Step 3: Find the tipping point.
Reaming $f'(x)=0$, solve this first-order differential equation about $x$ to find possible extreme points. Since this is an equation that contains exponential, trigonometric and polynomial terms, it is generally necessary to find its root numerically or in combination with graphical analysis.
Step 4: Determine the extreme value.
For each critical point $xi$ found, $f is calculated''(x)$ to determine whether they are extreme points and the corresponding extreme properties. The second derivative of the function $f(x)$ is: .
f''(x)=e^x(sin(x)2cos(x))6x$$。
If $f is satisfied at a certain critical point $xi$''(xi) >0$, then $xi$ is a local minima; If $f''(xi) <0$, then $xi$ is a local maximum point.
Step 5: Calculate and analyze the actual situation.
The exact location of the critical point is calculated in practice, which can involve complex algebraic operations and analytical skills, as well as mathematical software. Since this problem is an extreme value problem on an open interval, it is theoretically necessary to check the trend of the function at infinity (i.e., $xrightarrowpminfty$), but since $f(x)$ tends to positive or negative infinity as $x$ tends to positive or negative infinity, the exponential term will dominate and cause the whole function to tend to positive and negative infinity, so there is no need to consider the extreme value case on the boundary.
Through the above steps, we can roughly describe the method and process of solving the extreme points of the function $f(x)=e xsin(x)x 32x$ in the defined domain $mathbb$. The location and properties of the specific extreme points need to be calculated by $f'(x)$ is equal to zero, and further analysis of the second derivative sign near these points is determined.