Question

Let f(0)=3, f(1)=7, f(2)=12, f(3)=16. Find f(2).​

Answer 1

Answer:

12

Step-by-step explanation:

because f(2)=12

this is simple

Answer 2

Given,

• f(0)=3

• f(1)=7

• f(2)=12

• f(3)=16

To find,

The value of f(2)

Solution,

We can conclude from the given data that the equation will be a cubic polynomial with four unknown variables. With four equations and four unknown variables, we will find the values for all the variables and finally compute f(2).

The general form of the cubic equation = $ax^{3} + bx^{2} + cx + d$

The value of f(0) is 3.

$a(0)^{3} + b(0)^{2} + c*0 + d = 3\\d = 3$--eq(1)

The value of f(1) is 7.

$a(1)^{3} + b(1)^{2} + c*1 + d = 7\\ a + b + c + d = 7$--eq(2)

The value of f(2) is 12.

$a(2)^{3} + b(2)^{2} + c*2 + d = 3\\8a + 4b + c + d = 12$--eq(3)

The value of f(3) is 16.

$a(3)^{3} + b(2)^{2} + c*3 + d = 16\\27a + 9b + c + d = 16$--eq(4)

Combining all the four equations, we get the value of a,b,c,d.

$a = \frac{-31}{2}\\b = \frac{103}{2} \\c = -32\\d = 3$

The function becomes = $f(x) = \frac{-31}{2}x^{3} + \frac{103}{2} x^{2} -32x + 3$

$f(2) = \frac{-31}{2}2^{3} + \frac{103}{2} 2^{2} -322 + 3 = 21$

Therefore, the value of f(2) is 21.