relative maximum justification using first derivative test
Check the final answer first, then review the worked steps.
Problem
relative maximum justification using first derivative test
Answer
By the First Derivative Test, $g$ has a relative maximum at $x = c$ because $g'(c) = 0$ and $g'(x)$ changes from positive to negative at $x = c$.
Step-by-step solution
- Identify the critical point: A relative maximum or minimum of a differentiable function $g$ occurs at a critical point where the derivative $g'(x) = 0$. Looking at the graph, the brown line representing $g'$ crosses the x-axis at $x = c$, meaning $g'(c) = 0$.
- Analyze the sign change of the derivative: According to the First Derivative Test, if $g'(x)$ changes from positive to negative at a critical point $x = c$, then $g$ has a relative maximum at $x = c$.
- Observe the graph of $g'$: To the left of $x = c$ (where $x < c$), the graph of $g'$ is above the x-axis, meaning $g'(x) > 0$. To the right of $x = c$ (where $x > c$), the graph of $g'$ is below the x-axis, meaning $g'(x) < 0$.
- Formulate the justification: Since $g'(c) = 0$ and $g'(x)$ changes sign from positive to negative at $x = c$, the function $g$ must have a relative maximum at $x = c$ by the First Derivative Test.