It is very important to identify that the given derivative function is “cubic” in shape, so the derivative is at least a polynomial of an odd degree. Now, identifying the extrema for $f(x)$ is a bit more tricky. Therefore, only $x = 1.5$ is a relative minimum for $f(x)$. it is clear that $f'(x)$ changes from negative to positive about $x = 1.5$, but $f'(x)$ does not change signs about $x = 0$. We can then apply the first derivative test to identify local extrema for $f(x)$. It is clear that the derivative is equal to zero at $x = 0$ and $x = 1.5$. We can use the given graph of the derivative of the function to identify critical points. This would also be clear if the graph of the function was analyzed.
Therefore, there do not exist absolute extrema for the function either. It is clear that $f(x)$ is not defined at $x = 0$, meaning that there exists a vertical asymptote at $x = 0$. However, it is also important to check for the values at discontinuities for the function.
We can evaluate these limits by direct substitution. We identify a functions end behavior using limits. We can then identify the function’s absolute maximum and minimum by identifying the values of the function at the ends of its domain, or its end behavior. This implies that $x = 0$ is a relative maximum for $f(x)$ and $x = 2$ is a relative minimum for $f(x)$. It is clear from the graph that $f'(x)$ changes from positive to negative about $x = 0$, and changes from negative to positive about $x = 2$. We will use the graph to identify how $f'(x)$ changes about the critical points. There are many ways to identify how $f'(x)$ changes across a critical value, including sign patterns, graphical analysis, or even just inspection. We can do this by using the first derivative test. Now that we know the critical points, we must identify what type of extrema, if any, results from the critical values. Then, by factoring, we can identify the roots of the equation. We must first take the derivative of the function and set it equal to zero to identify the critical points of the function. Recall that the limit definition of the derivative is as follows.The correct answer is (D).
0 kommentar(er)
