The function (f(x)=ax 2bxc) is known, where (a,b,c) is a constant, and (aneq0). The function has a minimum at (x=1) and satisfies the conditions (f(0)=4) and (f(2)=f(-1)). Find the expression of the function (f(x)).
Ideas for solving the problem. Step 1: Understand the conditions of the question.
We need to understand what the condition given in the question means for the function (f(x)). The title tells us that (f(x)) is a quadratic function, whose graph is a parabola with an opening up or down (because (aneq0)) and a minimum at (x=1). This means that the abscissa of the vertex of the parabola is 1. (f(0)=4) tells us that when (x=0), the value of the function is 4. Then (f(2)=f(-1)) means that at (x=2) and (x=-1), the function values are equal.
Step 2: Apply an extreme condition.
Because (f(x)) has a minimum at (x=1), (f'(1)=0)。For (f(x)), we get (f'(x)=2axb)。Substituting (x=1) gives (2ab=0).
Step 3: Apply (f(0)=4).
Substituting (x=0) into (f(x)) gives (c=4). This is because (f(0)=acdot0 2bcdot0c=4).
Step 4: Apply (f(2)=f(-1)).
Substituting (x=2) and (x=-1) into (f(x)) respectively, we get two equations: .
1、(f(2)=4a2bc)
2、(f(-1)=abc)
Since (f(2)=f(-1)), we can set these two equations as an equation: (4a2bc=abc). Since we know (c=4), we can subtract (c) from the equation to get (3a3b=0).
Step 5: Solve the system of equations.
Now we have two equations:
1、(2ab=0)
2、(3a3b=0)
This is a system of linear equations containing two unknowns, (a) and (b). We can solve this system of equations using the elimination method or the substitution method. Using the elimination method, we can multiply the first equation by (3) and the second equation by (2) to get:
1、(6a3b=0)
2、(6a6b=0)
Subtract the first equation from the second equation to get (3b=0), so (b=0). Substituting (b=0) into any equation, e.g. (2ab=0), gives (2a=0), and therefore (a=0). But according to the problem condition, (aneq0), it means that we made a mistake in the calculation.