4) Completely Analyze \( K(t)=\left|3 t^{2}-14 t-5\right| \) Reason algebraically, i.e., do not refer to the following graph for your reasoning. Note: \( K(t)=\left\{\begin{array}{cl}3 t^{2}-14 t-5 & 3 t^{2}-14 t-5 \geq 0 \\ -\left(3 t^{2}-14 t-5\right) & 3 t^{2}-14 t-5<0\end{array}\right. \) This will help you get the derivative.

To completely analyze the function \( K(t)=\left|3 t^{2}-14 t-5\right| \), we need to consider the two cases separately: 1. When \( 3t^2 - 14t - 5 \geq 0 \): In this case, the absolute value function is equal to the expression inside the absolute value. So, \( K(t) = 3t^2 - 14t - 5 \) when \( 3t^2 - 14t - 5 \geq 0 \). 2. When \( 3t^2 - 14t - 5 < 0 \): In this case, the absolute value function is equal to the negative of the expression inside the absolute value. So, \( K(t) = -(3t^2 - 14t - 5) = -3t^2 + 14t + 5 \) when \( 3t^2 - 14t - 5 < 0 \). Now, let's find the derivative of the function \( K(t) \) with respect to \( t \) for each case: 1. For \( 3t^2 - 14t - 5 \geq 0 \): The derivative of \( 3t^2 - 14t - 5 \) with respect to \( t \) is \( 6t - 14 \). 2. For \( 3t^2 - 14t - 5 < 0 \): The derivative of \( -3t^2 + 14t + 5 \) with respect to \( t \) is \( -6t + 14 \). Therefore, the derivative of the function \( K(t) \) is: \[ K'(t) = \begin{cases} 6t - 14 & \text{if } 3t^2 - 14t - 5 \geq 0 \\ -6t + 14 & \text{if } 3t^2 - 14t - 5 < 0 \end{cases} \]