Step 1 :Take the first derivative of \(f(x)\) to find \(f^{\prime}(x)\): \(f^{\prime}(x)=3x^{2}-6x-24\)
Step 2 :Take the second derivative of \(f(x)\) to find \(f^{\prime \prime}(x)\): \(f^{\prime \prime}(x)=6x-6\)
Step 3 :Find the critical points of \(f\) by setting \(f^{\prime}(x)=0\): \(3x^{2}-6x-24=0\)
Step 4 :Simplify the equation to find the critical points: \(x^{2}-2x-8=0\)
Step 5 :Factor the equation to find the critical points: \((x-4)(x+2)=0\)
Step 6 :Set each factor equal to zero to find the critical points: \(x_{1}=4\) and \(x_{2}=-2\)
Step 7 :Find the inflection points of \(f\) by setting \(f^{\prime \prime}(x)=0\): \(6x-6=0\)
Step 8 :Solve for \(x\) to find the inflection point: \(x=1\)
Step 9 :Evaluate \(f\) at its critical points and at the endpoints of the given interval to identify local and global maxima and minima: \(f(4)=-33\), \(f(-2)=27\), \(f(-6)=-1\), and \(f(5)=-39\)
Step 10 :The local maximum is at \(x=-2\) with a value of \(27\), the local minimum is at \(x=4\) with a value of \(-33\), the global maximum is at \(x=-6\) with a value of \(-1\), and the global minimum is at \(x=5\) with a value of \(-39\)