Given the function $f(x)=\frac{1}{2} x^{4}+7 x^{3}-24 x^{2}$, find all intervals on which $f$ is concave down.

Step 1 ：Given the function \(f(x)=\frac{1}{2} x^{4}+7 x^{3}-24 x^{2}\), find all intervals on which \(f\) is concave down.

Step 2 ：First, find the first derivative of the function: \(f'(x) = 2x^3 + 21x^2 - 48x\)

Step 3 ：Next, find the second derivative of the function: \(f''(x) = 6x^2 + 42x - 48\)

Step 4 ：Find the critical points of the second derivative by setting it equal to zero: \(f''(x) = 0\)

Step 5 ：Critical points are: \(-8\) and \(1\)

Step 6 ：Test the intervals \((-\infty, -8)\), \((-8, 1)\), and \((1, \infty)\) to determine where the second derivative is negative.

Step 7 ：\(f(x)\) is concave down on the interval \(\boxed{(0, 1)}\)