predicting key aspects using a table\nx -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich…

predicting key aspects using a table\nx -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich interval contains a local maximum for this function?\nwhich interval contains a local minimum for this function?

predicting key aspects using a table\nx -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich interval contains a local maximum for this function?\nwhich interval contains a local minimum for this function?

Answer

Answer:

  • Local maximum interval: $(-1,1)$
  • Local minimum interval: $(-4,-1)$

Explanation:

Step1: Understand local maximum

A local maximum occurs when the function value is greater than the values at nearby points.

Step2: Analyze the table

We see that $f(-1) = 0$, $f(0)= - 2$, $f(1)=-4$. Since $f(-1)$ is greater than $f(0)$ and $f(1)$ in its neighborhood, a local - maximum is in the interval $(-1,1)$.

Step3: Understand local minimum

A local minimum occurs when the function value is less than the values at nearby points.

Step4: Analyze the table for local minimum

We have $f(-4)=-54$, $f(-3)= - 20$, $f(-2)=-4$, $f(-1)=0$. Since $f(-4)$ is less than $f(-3)$ and $f(-2)$ in its neighborhood, a local - minimum is in the interval $(-4,-1)$.