Step 1

Given that,

(1) \(\displaystyle{f}{''}{\left({x}\right)}={\cos{{x}}}\)

(2) PKf'(0)=3ZSK

(3) \(\displaystyle{f{{\left(\pi\right)}}}={5}\)

Integrate equation 1;

\(\displaystyle\int{f}\text{}{x}{)}{\left.{d}{x}\right.}=\int{\cos{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle{f}'{\left({x}\right)}={\sin{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}+{c}\) (4)

Using criteria (2);

\(\displaystyle{f}'{\left({0}\right)}={\sin{{\left({0}\right)}}}+{c}\)

\(\displaystyle{3}={0}+{c}\)

\(\displaystyle\Rightarrow{c}={3}\)

Equation (4) is are written as,

\(\displaystyle{f}'{\left({x}\right)}={\sin{{\left({x}\right)}}}+{3}\) ..(5)

Step 2

Integrate equation 5;

\(\displaystyle\int{f}'{\left({x}\right)}{\left.{d}{x}\right.}=\int{\left[{\sin{{\left({x}\right)}}}+{3}\right]}{\left.{d}{x}\right.}\)

\(\displaystyle{f{{\left({x}\right)}}}=-{\cos{{\left({x}\right)}}}+{3}{x}+{c}'\) (5)

Using criteria (3),

\(\displaystyle{f{{\left(\pi\right)}}}=-{\cos{{\left(\pi\right)}}}+{3}\pi+{c}'\)

\(\displaystyle{5}={1}+{3}\pi+{c}'\)

\(\displaystyle\Rightarrow{c}'={4}-{3}\pi\)

Equation (5) is are written as,

\(\displaystyle{f{{\left({x}\right)}}}=-{\cos{{\left({x}\right)}}}+{3}{x}+{4}-{3}\pi\)

Given that,

(1) \(\displaystyle{f}{''}{\left({x}\right)}={\cos{{x}}}\)

(2) PKf'(0)=3ZSK

(3) \(\displaystyle{f{{\left(\pi\right)}}}={5}\)

Integrate equation 1;

\(\displaystyle\int{f}\text{}{x}{)}{\left.{d}{x}\right.}=\int{\cos{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle{f}'{\left({x}\right)}={\sin{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}+{c}\) (4)

Using criteria (2);

\(\displaystyle{f}'{\left({0}\right)}={\sin{{\left({0}\right)}}}+{c}\)

\(\displaystyle{3}={0}+{c}\)

\(\displaystyle\Rightarrow{c}={3}\)

Equation (4) is are written as,

\(\displaystyle{f}'{\left({x}\right)}={\sin{{\left({x}\right)}}}+{3}\) ..(5)

Step 2

Integrate equation 5;

\(\displaystyle\int{f}'{\left({x}\right)}{\left.{d}{x}\right.}=\int{\left[{\sin{{\left({x}\right)}}}+{3}\right]}{\left.{d}{x}\right.}\)

\(\displaystyle{f{{\left({x}\right)}}}=-{\cos{{\left({x}\right)}}}+{3}{x}+{c}'\) (5)

Using criteria (3),

\(\displaystyle{f{{\left(\pi\right)}}}=-{\cos{{\left(\pi\right)}}}+{3}\pi+{c}'\)

\(\displaystyle{5}={1}+{3}\pi+{c}'\)

\(\displaystyle\Rightarrow{c}'={4}-{3}\pi\)

Equation (5) is are written as,

\(\displaystyle{f{{\left({x}\right)}}}=-{\cos{{\left({x}\right)}}}+{3}{x}+{4}-{3}\pi\)