For the following function, find the intervals on which the function is increasing or decreasing. \( f(x)=2 x^{3}+3 x^{2}-12 x+13 \)

To find the intervals where the function \( f(x)=2 x^{3}+3 x^{2}-12 x+13 \) is increasing or decreasing, we first need to compute the derivative: \( f'(x) = 6x^2 + 6x - 12 \). Next, we set the derivative equal to zero to find the critical points: \( 6x^2 + 6x - 12 = 0 \) Simplifying gives us: \( x^2 + x - 2 = 0 \) Factoring this quadratic equation yields: \( (x + 2)(x - 1) = 0 \) Thus, \( x = -2 \) and \( x = 1 \) are critical points. Now, we test the intervals around these points: - For \( x < -2 \) (e.g., \( x = -3 \)), \( f'(-3) = 6(-3)^2 + 6(-3) - 12 > 0 \) (increasing). - For \( -2 < x < 1 \) (e.g., \( x = 0 \)), \( f'(0) = 6(0)^2 + 6(0) - 12 < 0 \) (decreasing). - For \( x > 1 \) (e.g., \( x = 2 \)), \( f'(2) = 6(2)^2 + 6(2) - 12 > 0 \) (increasing). So, the function is increasing on \( (-\infty, -2) \) and \( (1, \infty) \), and decreasing on \( (-2, 1) \). Happy graphing!