question 9\n5 pts\n$f$ is continuous on $(-\\infty,\\infty)$. if $f(a)=0$ and $f(a)>0$, which of the…

question 9\n5 pts\n$f$ is continuous on $(-\\infty,\\infty)$. if $f(a)=0$ and $f(a)>0$, which of the following is correct?\n$f$ has a local minimum at $x = a$.\n$f$ has neither a maximum nor a minimum at $x = a$.\nmore information is needed to determine if $f$ has a maximum or a minimum at $x = a$.\n$f$ has a local maximum at $x = a$.

question 9\n5 pts\n$f$ is continuous on $(-\\infty,\\infty)$. if $f(a)=0$ and $f(a)>0$, which of the following is correct?\n$f$ has a local minimum at $x = a$.\n$f$ has neither a maximum nor a minimum at $x = a$.\nmore information is needed to determine if $f$ has a maximum or a minimum at $x = a$.\n$f$ has a local maximum at $x = a$.

Answer

Explanation:

Step1: Recall the second - derivative test

The second - derivative test states that if (f^{\prime}(a)=0) (so (x = a) is a critical point) and (f^{\prime\prime}(x)) is continuous at (x=a).

Step2: Apply the second - derivative test

If (f^{\prime}(a) = 0) and (f^{\prime\prime}(a)>0), then the function (f(x)) has a local minimum at (x = a). This is because when (f^{\prime\prime}(a)>0), the graph of (y = f(x)) is concave up at (x=a). At a critical point ((f^{\prime}(a) = 0)) with concave - up shape, the function has a local minimum.

Answer:

(f) has a local minimum at (x = a).