Question

find f.

$f(x)=-2 + 24x - 12x^{2}$, $f(0)=6$, $f(0)=14$

$f(x)=$

Explanation:

Step1: Integrate \(f''(x)\) to find \(f'(x)\)

We know that if \(f''(x)=-2 + 24x-12x^{2}\), then by the power rule of integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)):

$$
LATEXBLOCK0
$$

Use the initial condition \(f'(0) = 14\). Substitute \(x = 0\) into \(f'(x)\):
\(f'(0)=-2\times0+12\times0^{2}-4\times0^{3}+C\), so \(C = 14\)
Then \(f'(x)=-4x^{3}+12x^{2}-2x + 14\)

Step2: Integrate \(f'(x)\) to find \(f(x)\)

$$
LATEXBLOCK1
$$

Use the initial condition \(f(0)=6\). Substitute \(x = 0\) into \(f(x)\):
\(f(0)=-0^{4}+4\times0^{3}-0^{2}+14\times0+D\), so \(D = 6\)

So \(f(x)=-x^{4}+4x^{3}-x^{2}+14x + 6\)

Answer:

\(f(x)=-x^{4}+4x^{3}-x^{2}+14x + 6\)