Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)=a\cos x+\cos 3x$

An extremum (local maximum or minimum) occurs at a point where the first derivative of the function is zero. We set

$f^{\prime }(x)=0$

$a\cos x+\cos 3x=0$

$x$ value and solve for $a$

We are given that the extremum occurs at $x=\frac{\pi }{3}$. Substituting this value into the derivative equation:

$a\cos (\frac{\pi }{3})+\cos (3\cdot \frac{\pi }{3})=0$

$a\cos (\frac{\pi }{3})+\cos (\pi )=0$

Using the known trigonometric values $\cos (\frac{\pi }{3})=\frac{1}{2}$ and $\cos (\pi )=-1$, we get:

$a(\frac{1}{2})+(-1)=0$

$\frac{a}{2}-1=0$

$\frac{a}{2}=1$