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已知函数 f(x)=∣ax2−∣x−2∣∣+x−af(x)=\left|a x^{2}-\right| x-2||+x-a ，若函数 f(x)f(x) 恰有 4 个零点，则实数 aa 的取值范围是
Mar 8, 2024
已知函数 f(x)=∣ax2−∣x−2∣∣+x−af(x)=\left|a x^{2}-\right| x-2||+x-a ，若函数 f(x)f(x) 恰有 4 个零点，则实数 aa 的取值范围是
Solution by Steps
step 1
To find the range of values for a a such that the function f(x)=∣ax2−x−2∣+x−a f(x) = |ax^2 - x - 2| + x - a has exactly 4 real roots, we need to analyze the behavior of the function and the absolute value expression
step 2
The absolute value expression ∣ax2−x−2∣ |ax^2 - x - 2| can be split into two cases depending on the sign of the expression inside the absolute value
step 3
For ax2−x−2≥0 ax^2 - x - 2 \geq 0 , the function simplifies to f(x)=ax2−x−2+x−a f(x) = ax^2 - x - 2 + x - a . For ax^2 - x - 2 < 0 , the function simplifies to f(x)=−(ax2−x−2)+x−a f(x) = -(ax^2 - x - 2) + x - a
step 4
Solving ax2−x−2=0 ax^2 - x - 2 = 0 gives us the points where the expression inside the absolute value changes sign
step 5
The solutions to ax2−x−2=0 ax^2 - x - 2 = 0 are x=−1 x = -1 and x=2a x = \frac{2}{a} if a≠0 a \neq 0 . These are the critical points for the absolute value expression
step 6
We need to find the values of a a such that there are 4 real roots, which means the graph of f(x) f(x) must intersect the x-axis at 4 points
step 7
According to the asksia-ll calculation list, the values of a a are given by the following expressions depending on the intervals for x x :
step 8
For x < -1 , a=2(x+1)x2+1 a = \frac{2(x + 1)}{x^2 + 1}
step 9
For −1≤x≤1 -1 \leq x \leq 1 , a=2(x+1)x2+1 a = \frac{2(x + 1)}{x^2 + 1}
step 10
For 1 < x < \frac{1}{3} \left(27 - 3 \sqrt{78}\right)^{\frac{1}{3}} + \frac{\left(9 + \sqrt{78}\right)^{\frac{1}{3}}}{3^{\frac{2}{3}}} , a=2(x+1)x2+1 a = \frac{2(x + 1)}{x^2 + 1}
step 11
By analyzing these expressions and intervals, we can determine the range of a a for which there are 4 real roots
Answer
The range of values for a a such that the function f(x) f(x) has exactly 4 real roots is not explicitly provided in the asksia-ll calculation list. Further analysis is required to determine the exact range.
Key Concept
Absolute Value Function Analysis
Explanation
To determine the number of real roots of a function involving an absolute value, we analyze the function by considering the different cases for the expression inside the absolute value. We then solve for the variable of interest, in this case a a , within the specified intervals to find the range of values that satisfy the condition of having a certain number of real roots.
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