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Let
f be a twice differentiable function, and let
f(-3)=-4,
f^(')(-3)=0, and
f^('')(-3)=1.
What occurs in the graph of
f at the point
(-3,-4) ?
Choose 1 answer:
(A)
(-3,-4) is a minimum point.
(B)
(-3,-4) is a maximum point.
(C) There's not enough information to tell.

Let $f$ be a twice differentiable function, and let $f(-3)=-4$ , $f^{\prime}(-3)=0$ , and $f^{\prime \prime}(-3)=1$ . $\newline$ What occurs in the graph of $f$ at the point $(-3,-4)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $(-3,-4)$ is a minimum point. $\newline$ (B) $(-3,-4)$ is a maximum point. $\newline$ (C) There's not enough information to tell.

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Full solution

Q. Let

f

be a twice differentiable function, and let

f(-3)=-4

,

f^{\prime}(-3)=0

, and

f^{\prime \prime}(-3)=1

.

\newline

What occurs in the graph of

f

at the point

(-3,-4)

?

\newline

Choose

1

answer:

\newline

(A)

(-3,-4)

is a minimum point.

\newline

(B)

(-3,-4)

is a maximum point.

\newline

(C) There's not enough information to tell.

Given Information: Given that $f$ is a twice differentiable function, we have information about the function and its first two derivatives at the point $x = -3$ . We know that $f(-3) = -4$ , $f'(-3) = 0$ , and $f''(-3) = 1$ .
Point on Graph: The value $f(-3) = -4$ tells us that the point $(-3, -4)$ lies on the graph of the function $f$ . However, this information alone does not tell us about the nature of the point on the graph.
Tangent Line: The derivative $f'(-3) = 0$ indicates that the slope of the tangent to the graph of $f$ at $x = -3$ is zero. This means that the graph of $f$ has a horizontal tangent line at the point $(-3, -4)$ . This could be indicative of a local maximum, a local minimum, or a point of inflection.
Concavity Analysis: The second derivative $f''(-3) = 1$ tells us about the concavity of the graph at $x = -3$ . Since $f''(-3)$ is positive, the graph of $f$ is concave up at this point. This suggests that the point $(-3, -4)$ is a local minimum.
Conclusion: Combining the information from the first and second derivatives, we can conclude that the point $(-3, -4)$ is a local minimum on the graph of $f$ because the first derivative is zero (indicating a horizontal tangent) and the second derivative is positive (indicating concave up).

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