find f. f(x) = 5 + cos(x), f(0) = -1, f(3π/2) = 0 f(x) =

Explanation:

Step1: Integrar $f''(x)$ para obtener $f'(x)$

$$f'(x)=\int(5 + \cos(x))dx=5x+\sin(x)+C_1$$

Step2: Integrar $f'(x)$ para obtener $f(x)$

$$f(x)=\int(5x+\sin(x)+C_1)dx=\frac{5}{2}x^{2}-\cos(x)+C_1x + C_2$$

Step3: Usar la condición $f(0)= - 1$

Sustituir $x = 0$ en $f(x)$: $$f(0)=\frac{5}{2}(0)^{2}-\cos(0)+C_1(0)+C_2=-1$$ $$-1 + C_2=-1\Rightarrow C_2 = 0$$

Step4: Usar la condición $f(\frac{3\pi}{2})=0$

Sustituir $x=\frac{3\pi}{2}$ y $C_2 = 0$ en $f(x)$: $$f(\frac{3\pi}{2})=\frac{5}{2}(\frac{3\pi}{2})^{2}-\cos(\frac{3\pi}{2})+C_1(\frac{3\pi}{2})+0 = 0$$ $$\frac{45\pi^{2}}{8}+0+\frac{3\pi}{2}C_1=0$$ $$\frac{3\pi}{2}C_1=-\frac{45\pi^{2}}{8}$$ $$C_1=-\frac{15\pi}{4}$$

Answer:

$f(x)=\frac{5}{2}x^{2}-\cos(x)-\frac{15\pi}{4}x$