Question

Find f such that the given conditions are satisfied. 

$$ f(x)=x^{2}-3 x+9, f(0)=5 $$

\(f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+9 x+5\)

\(f(x)=\frac{1}{3} x^{3}-4 x^{2}+9 x+5\)

\(f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+9 x+1\)

\(f(x)=\frac{1}{3} x^{3}-4 x^{2}+9 x+1\)

Answer 1

Let, f'(x) = x² - 3x + 9

integrate both sides with respect to 'x',

    ∫f'(x) = ∫(x² - 3x + 9)dx

=>f(x) = ∫(x²)dx - 3∫(x)dx + 9∫dx

=>f(x) = x³/3 - (3/2)x² + 9x + C

if f(0) = 5, then,

(0)³/3 - (3/2)(0)² + 9(0) + C = 5

=> C = 5

therefore,  f(x) = x³/3 - (3/2)x² + 9x + C

the first option is your answer.

Answer 2

f'in = 22-32 +9 flol=5 Jf'(de = 5x2-32+91 du Hu) = -2 *+9*+. New flol=5 5=oto toto c=5 2 +92 +5 A 3 x flas = 73 3