Respuesta :
Answer:
Therefore the required solution is
[tex]f(x)=4x^2-2x+5[/tex]
Step-by-step explanation:
Rule of integration:
- [tex]\int x^ndx=\frac{x^{n+1}}{n+1}+C[/tex]
- [tex]\int m\ dx = m\int dx= mx+C[/tex] [ m is a constant]
- [tex]\int mx^ndx=m\int x^n dx=m.\frac{x^{n+1}}{n+1}[/tex]
Given that,
f''(x) = 8 and initial conditions are f'(1)=6 and f(0)=5
∴f''(x) = 8
Integrating both sides
[tex]\int f''(x) dx=\int 8 dx[/tex]
[tex]\Rightarrow f'(x)= 8x+C_1[/tex] [ [tex]C_1[/tex] is constant of integration]
Initial condition f'(1) =6
[tex]\therefore 6= 8.1+C_1[/tex]
[tex]\Rightarrow C_1=6-8[/tex]
[tex]\Rightarrow C_1=-2[/tex]
[tex]\therefore f'(x)=8x-2[/tex]
Again integrating both sides
[tex]\int f'(x) dx=\int8x dx- \int2dx[/tex]
[tex]\Rightarrow f(x)=8\frac{x^2}{2}-2x+C_2[/tex] [ [tex]C_2[/tex] is constant of integration]
[tex]\Rightarrow f(x)=4x^2-2x+C_2[/tex]
Initial condition f(0)=5
[tex]5=4.0^2-2.0+C_2[/tex]
[tex]\Rightarrow C_2=5[/tex]
[tex]\therefore f(x)=4x^2-2x+5[/tex]
Therefore the required solution is
[tex]f(x)=4x^2-2x+5[/tex]
