Questions: Given the function f(x)=4 x^4-24 x^2, determine all intervals on which f is both increasing and concave down.

Determine the intervals on which the function \( f(x) = 4x^4 - 24x^2 \) is both increasing and concave down.

Find the first derivative \( f'(x) \).

The first derivative is calculated as \( f'(x) = 16x^3 - 48x \).

Find the second derivative \( f''(x) \).

The second derivative is calculated as \( f''(x) = 48x^2 - 48 \).

Identify the critical points by solving \( f'(x) = 0 \).

The critical points are found to be \( x = 0, -\sqrt{3}, \sqrt{3} \).

Identify the inflection points by solving \( f''(x) = 0 \).

The inflection points are found to be \( x = -1, 1 \).

Analyze the intervals determined by the critical and inflection points.

The intervals to analyze are \( (-\sqrt{3}, -1), (-1, 0), (0, 1), (1, \sqrt{3}) \). The analysis shows that the function is increasing and concave down in the interval \( (-1, 0) \).

The interval on which \( f \) is both increasing and concave down is \( \boxed{(-1, 0)} \).

The interval on which \( f \) is both increasing and concave down is \( \boxed{(-1, 0)} \).