A cubic polynomial function fx has leading coefficient 2 and

Solution

Let the expression of the cubic polynomial function f(x) be y = f (x) = -2x³ + bx² + cx + d ...(1)

Since f (x) intercepts the Y -axis at 2, therefore, on substituting x = 0 in the above equation, we get 2 = d. Now, on substituting d = 2 in the equation above, the expression for f (x) changes to y = f(x) = -2x³ + bx² + cx + 2...(2)

Since f ( 1 ) = 1, on substituting x = 1 in the 2nd equation, we get -2(1)³ + b(1)² + c (1) + 2 = 1or, b + c = 1 - 2 + 2 or, b + c = 1...(3).

Also, since f ( - 2 ) = - 2, on substituting x = - 2 in the 2nd equation, we get -2(-2)³ + b( - 2)² + c (-2) + 2 = - 2 or, 4b - 2c = - 2 - 16 - 2 or, 4b - 2c = -20 or, 2b - c = -10 ...(4).

On adding the 3rd and the 4th equations, we get b + c + 2b - c = 1 - 10 or, 3b = - 9 so that b = - 3. On substituting b = - 3 in the 3rd equation, we get - 3 + c = 1 or, c = 1 + 3 = 4.

On substituting b = -3 and c = 4 in the 2nd equation, we get f (x) = -2x³ -3x² + 4x + 2. We can verify the result by ckecking the values of f(1) and f ( -2). On substituting x = 1, we get f (1) = -2 -3 + 4 + 2 = 1 . Also, on substituting x = -2, we get f (-2) = 16 -12- 8 + 2 = -2. Thus we have derived the expression for f (x) correctly. Now, on substituting x = -1, we get f ( - 1) = - 2( - 1)³ -3( - 1)² +4( - 1) + 2 = 2 - 3 - 4 + 2 = - 3. The answer is

f(x) =  -2x³ -3x² + 4x + 2

f(-1) = -3