(Solved): EXAMPLE 2 Find the local minimum and maximum values of the function below. \[ f(x)=3 x^{4}-16 x^{3} ...

EXAMPLE 2 Find the local minimum and maximum values of the function below. \[ f(x)=3 x^{4}-16 x^{3}-126 x^{2}+5 \] Video Example SOLUTION \( f^{\prime}(x)=12 x^{3}-48 x^{2}-252 x=12 x(x-7)(x+3) \) From the chart we see that \( f^{\prime}(x) \) changes from negative to positive at \( -3 \), so \( f(-3)=\quad \times \) is a local minimum value by the First Derivative Test. Similarly, \( f^{\prime}(x) \) changes from negative to positive at 7 , so \( f(7)= \) is a local minimum value. And, \( f(0)= \) \( x \) is a local maximum value because \( f^{\prime}(x) \) changes from positive to negative at 0 .