Knowing the function $f(x)=frac$, find the extreme point of this function in the defined domain.
Solution: For the extreme value problem of such fractional functions, we can solve it by finding the derivative. But before we can do that, we need to determine the domain in which the function is defined. Observing the numerator and denominator, it is obvious that the function only makes sense if the denominator $x 22$ is not zero, and since $x 22$ is everybody at zero, the domain of the function $f(x)$ is the whole real number $mathbb$.
Step 1: Simplify or decompose.
Try to factor or simplify the function, but the polynomial in this problem is not easy to be factored directly, so we can go directly to the next step - derivation.
Step 2: Derivative.
Set $f'(x)$ is a derivative of $f(x)$, then there is:
f'(x)=frac$$。
Applying the power function and chain rule, the following is calculated:
f'(x)=frac$$。
Step 3: Find the tipping point.
Reaming $f'(x)=0$, solve this cubic equation to find possible extreme points:
4x^36x^210x10=0$$
Due to the complexity of the equation, it can be solved numerically or mathematically as $x1, x2, x3$ (assuming there are three real roots).
Step 4: Determine the extreme value.
At the found critical points $x1, x2, x3$, the second derivative $f is further calculated''(x)$ to determine whether the points are extreme points and whether they correspond to maximums or minimums. Calculate the second derivative to yield:
f''(x)=frac$$。
The $f at each critical point is then calculated separately''(x)$ value, according to the second derivative discriminant method to determine the extreme value property: if $f''(xi) >0$, then $xi$ is the minimum point; If $f''(xi) <0$, then $xi$ is the maximum point.
In the above steps, we can find all the extreme points of the function $f(x)=frac$ in the defined domain and determine their corresponding extreme types. However, since the specific critical point needs to be obtained by solving the cubic equation, no specific numerical results are given here.