Question

If f(x) is a quadratic polynomial such that f(0)=3, f'(2)=2 ,and f'(3)=12 then find f(x)

Answer 1

f(x) = 5x² - 18x + 3.

Step-by-step explanation:

f(x) is a quadratic polynomial.

Let f(x) = ax² + bx + c....................(i)

f(0) = 3

=> a(0)² + b(0) + c = 3

=> c = 3

Differentiate equation(i) with respect to x,

f'(x) = 2ax + b

f'(2) = 2

=> 2a(2) + b = 2

=> 4a + b = 2.....................(ii)

Now, f'(3) = 12

=> 2a(3) + b = 12

=> 6a + b = 12...................(iii)

Subtract equation(ii) from equation(iii),

6a + b - 4a - b = 12 - 2

=> 2a = 10

=> a = $\frac{10}{2}$ = 5

Plug the value of a in equation(i),

=> 4(5) + b = 2

=> 20 + b = 2

=> b = 2 - 20 = -18

Plug the value of a, b and c in equation(i),

f(x) = 5x² - 18x + 3

Hence, f(x) = 5x² - 18x + 3.