given the sign chart for the derivative f(x) below, choose the x -value where the function f(x) has a…

given the sign chart for the derivative f(x) below, choose the x -value where the function f(x) has a relative minimum. f(x) + + + + 0 - - - 0 - - - 0 + + + + 0 1 2 3 4 x=1 x=2 x=4 x=0 x=3 none of the above

given the sign chart for the derivative f(x) below, choose the x -value where the function f(x) has a relative minimum. f(x) + + + + 0 - - - 0 - - - 0 + + + + 0 1 2 3 4 x=1 x=2 x=4 x=0 x=3 none of the above

Answer

Explanation:

Step1: Recall the first - derivative test

A function $y = f(x)$ has a relative minimum at a point $c$ if $f^{\prime}(x)$ changes sign from negative to positive at $x = c$.

Step2: Analyze the sign - chart of $f^{\prime}(x)$

Looking at the sign - chart of $f^{\prime}(x)$:

  • Before $x = 1$, $f^{\prime}(x)>0$ and after $x = 1$, $f^{\prime}(x)<0$, so $x = 1$ is a relative maximum.
  • Before $x = 2$, $f^{\prime}(x)<0$ and after $x = 2$, $f^{\prime}(x)<0$, so $x = 2$ is neither a relative maximum nor a relative minimum.
  • Before $x = 3$, $f^{\prime}(x)<0$ and after $x = 3$, $f^{\prime}(x)>0$.

Answer:

$x = 3$