Question

Given that f ′′ ( x ) = cos ( x ) f ″ ( x ) = cos ⁡ ( x ) , f ′ ( π / 2 ) = 7 f ′ ( π / 2 ) = 7 and f ( π / 2 ) = 5 f ( π / 2 ) = 5 find: f ′ ( x ) = f ′ ( x ) = f ( x ) = f ( x ) =

Answer 1

Answer:
`\sin x+6`

`-\cos x+6x+5-3\pi`

Explanation:

Given

`f''(x)=\cos x`

Integrating the equation

`\Rightarrow f''(x)=(d^2y)/(dx^2)=\cos x\\\\\Rightarrow \int \frac{\mathrm{d^2} y}{\mathrm{d} x^2}=\int \cos x\\\\\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=\sin x+c`

Put
`x=(\pi)/(2)\ \text{and}\ (dy)/(dx)=7`

`\Rightarrow 7=\sin (\pi )/(2)+c\\\Rightarrow c=7-1\\\Rightarrow c=6`

`\Rightarrow (dy)/(dx)=\sin x+6`

again integrating both sides of the equation

`\Rightarrow \int \frac{\mathrm{d} y}{\mathrm{d} x}=\int (\sin x+6)dx\\\\\Rightarrow y=-\cos x+6x+c_1`

Put
`x=(\pi)/(2)\ \text{and}\ y=5`

`\Rightarrow 5=-\cos (\pi)/(2)+6((\pi)/(2))+c_1\\\Rightarrow c_1=5-3\pi`

`\therefore f(x)=y=-\cos x+6x+5-3\pi`