Let f be a twice differentiable function, and let f(3)=8, f^(')(3)=0, and f^('')(3)=0. What occurs in the graph of f at the point (3,8) ? Choose 1 answer: (A) (3,8) is a minimum point. (B) (3,8) is a maximum point. (C) There's not enough information to tell.

Given Information: To determine what occurs at the point (3,8), we need to analyze the given information about the function f and its derivatives at x = 3. Given: f(3) = 8, which means the point (3,8) lies on the graph of f. f'(3) = 0, which indicates that the slope of the tangent to the graph of f at x = 3 is zero. This could mean that (3,8) is a maximum, a minimum, or a point of inflection. f''(3) = 0, which means the concavity of the graph at x = 3 is neither upwards nor downwards. This does not provide conclusive information about whether (3,8) is a maximum or minimum point.Analysis of f'(x): To determine if (3,8) is a maximum or minimum point, we would need the sign of the second derivative around x = 3. Since f''(3) = 0, we cannot conclude whether the graph is concave up or concave down at this point. If f''(x) were positive for x near 3, it would indicate a minimum point. If f''(x) were negative for x near 3, it would indicate a maximum point. However, with f''(3) = 0, we cannot make this determination.Determining Maximum or Minimum: Since we do not have information about the behavior of f''(x) around x = 3 other than at the exact point x = 3, we cannot determine if (3,8) is a maximum or minimum point. We need additional information about the concavity of f near x = 3 to make a definitive conclusion.

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Let
f be a twice differentiable function, and let
f(3)=8,
f^(')(3)=0, and
f^('')(3)=0.
What occurs in the graph of
f at the point
(3,8) ?
Choose 1 answer:
(A)
(3,8) is a minimum point.
(B)
(3,8) is a maximum point.
(C) There's not enough information to tell.

Let $f$ be a twice differentiable function, and let $f(3)=8$ , $f^{\prime}(3)=0$ , and $f^{\prime \prime}(3)=0$ . $\newline$ What occurs in the graph of $f$ at the point $(3,8)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $(3,8)$ is a minimum point. $\newline$ (B) $(3,8)$ is a maximum point. $\newline$ (C) There's not enough information to tell.

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Grade 7

One-step inequalities: word problems

Full solution

Q. Let

f

be a twice differentiable function, and let

f(3)=8

f^{\prime}(3)=0

, and

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

\newline

What occurs in the graph of

f

at the point

(3,8)

\newline

Choose

1

answer:

\newline

(A)

(3,8)

is a minimum point.

\newline

(B)

(3,8)

is a maximum point.

\newline

Given Information: To determine what occurs at the point $(3,8)$ , we need to analyze the given information about the function $f$ and its derivatives at $x = 3$ . $\newline$ Given: $\newline$ $f(3) = 8$ , which means the point $(3,8)$ lies on the graph of $f$ . $\newline$ $f'(3) = 0$ , which indicates that the slope of the tangent to the graph of $f$ at $x = 3$ is zero. This could mean that $(3,8)$ is a maximum, a minimum, or a point of inflection. $\newline$ $f$ $0$ , which means the concavity of the graph at $x = 3$ is neither upwards nor downwards. This does not provide conclusive information about whether $(3,8)$ is a maximum or minimum point.
Analysis of $f'(x)$ : To determine if $(3,8)$ is a maximum or minimum point, we would need the sign of the second derivative around $x = 3$ . Since $f''(3) = 0$ , we cannot conclude whether the graph is concave up or concave down at this point. If $f''(x)$ were positive for $x$ near $3$ , it would indicate a minimum point. If $f''(x)$ were negative for $x$ near $3$ , it would indicate a maximum point. However, with $f''(3) = 0$ , we cannot make this determination.
Determining Maximum or Minimum: Since we do not have information about the behavior of $f''(x)$ around $x = 3$ other than at the exact point $x = 3$ , we cannot determine if $(3,8)$ is a maximum or minimum point. We need additional information about the concavity of $f$ near $x = 3$ to make a definitive conclusion.