Question

If $f\left(x\right)=k(\cos x-\sin x),f\left(0\right)=3,f^{\prime }\left(\frac{\pi }{2}\right)=15$, find $f\left(x\right)$.

Answer 1

We have,

$\begin{array}{c}{f}^{\prime }\left(x\right)=k\left(\cos x-\sin x\right)........given\\ f\left(x\right)=\int f^{\prime }\left(x\right)dx\\ =\int k\left(\cos x-\sin x\right)dx\\ =k\int \left(\cos x-\sin x\right)dx\\ f\left(x\right)=k\left(\sin x+\cos x\right)+C......\left(i\right)\\ f\left(0\right)=\mathrm{3........}\left(given\right)\\ k\left(\cos 0-\sin 0\right)=3\\ k\left(1\right)=3\\ k=\mathrm{3.......}\left(ii\right)\\ also,f^{\prime }\left(\frac{\pi }{2}\right)=15\\ k\left[\sin \left(\frac{\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)\right]+c=15\\ 3\left(1+0\right)+c=15\\ c=12\\ putting\left(ii\right)and\left(iii\right)in\left(i\right).weget\\ f\left(x\right)=3\left(\sin x+\cos x\right)+12\end{array}$