Find an expression for a cubic function [tex]f[/tex] if [tex]f(5) = 100[/tex] and [tex]f(-5) = f(0) = f(6) = 0[/tex].



Answer :

Sure, let's find an expression for the cubic function [tex]\( f(x) \)[/tex] given the conditions [tex]\( f(5) = 100 \)[/tex] and [tex]\( f(-5) = f(0) = f(6) = 0 \)[/tex].

### Step-by-Step Solution

1. Identify the Roots:

Since [tex]\( f(-5) = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex], and [tex]\( f(6) = 0 \)[/tex], these points (-5, 0, 6) are the roots of the cubic function. Hence, we can express the cubic function as:

[tex]\[ f(x) = a \cdot (x - r_1) \cdot (x - r_2) \cdot (x - r_3) \][/tex]

where [tex]\( r_1, r_2, \)[/tex] and [tex]\( r_3 \)[/tex] are the roots of the function.

2. Substitute the Roots:

Given the roots, we have [tex]\( r_1 = -5 \)[/tex], [tex]\( r_2 = 0 \)[/tex], and [tex]\( r_3 = 6 \)[/tex]. Thus, the cubic function becomes:

[tex]\[ f(x) = a \cdot (x + 5) \cdot x \cdot (x - 6) \][/tex]

3. Determine the Coefficient [tex]\( a \)[/tex]:

We are given [tex]\( f(5) = 100 \)[/tex]. Using this information, we can solve for the coefficient [tex]\( a \)[/tex]:

[tex]\[ f(5) = a \cdot (5 + 5) \cdot 5 \cdot (5 - 6) \][/tex]

Simplify the expression:

[tex]\[ 100 = a \cdot 10 \cdot 5 \cdot (-1) \][/tex]

[tex]\[ 100 = -50a \][/tex]

Solving for [tex]\( a \)[/tex]:

[tex]\[ a = -2 \][/tex]

4. Formulate the Final Function:

Now that we have the value of [tex]\( a \)[/tex], we can write the final cubic function as:

[tex]\[ f(x) = -2 \cdot (x + 5) \cdot x \cdot (x - 6) \][/tex]

### Conclusion

The cubic function that satisfies the given conditions [tex]\( f(5) = 100 \)[/tex] and [tex]\( f(-5) = f(0) = f(6) = 0 \)[/tex] is:

[tex]\[ f(x) = -2 \cdot (x + 5) \cdot x \cdot (x - 6) \][/tex]

This is the required cubic function.