Let f be a twice differentiable function, and let f(2)=3, f^(')(2)=0, and f^('')(2)=5. What occurs in the graph of f at the point (2,3) ? Choose 1 answer: (A) (2,3) is a minimum point. (B) (2,3) is a maximum point. (C) There's not enough information to tell.

Given Information: We are given that f is a twice differentiable function, which means it is smooth and has a continuous first and second derivative. We are also given that f(2) = 3, f'(2) = 0, and f''(2) = 5. The value f(2) = 3 tells us that the point (2,3) lies on the graph of f. The value f'(2) = 0 indicates that the slope of the tangent to the graph of f at x = 2 is zero, which means the graph has a horizontal tangent line at this point. This could be indicative of a local maximum, local minimum, or a point of inflection.Analysis of f'(2): To determine whether (2,3) is a maximum, minimum, or neither, we need to consider the second derivative, f''(2). The value f''(2) = 5 is positive, which tells us that the concavity of the graph at x = 2 is upwards (like a smile). According to the second derivative test, if the first derivative at a point is zero and the second derivative at that point is positive, then the function has a local minimum at that point.Second Derivative Test: Since f'(2) = 0 and f''(2) > 0, we can conclude that (2,3) is a local minimum point on the graph of f. This answers the question prompt.

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Let
f be a twice differentiable function, and let
f(2)=3,
f^(')(2)=0, and
f^('')(2)=5.
What occurs in the graph of
f at the point
(2,3) ?
Choose 1 answer:
(A)
(2,3) is a minimum point.
(B)
(2,3) is a maximum point.
(C) There's not enough information to tell.

Let $f$ be a twice differentiable function, and let $f(2)=3$ , $f^{\prime}(2)=0$ , and $f^{\prime \prime}(2)=5$ . $\newline$ What occurs in the graph of $f$ at the point $(2,3)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $(2,3)$ is a minimum point. $\newline$ (B) $(2,3)$ 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(2)=3

f^{\prime}(2)=0

, and

f^{\prime \prime}(2)=5

\newline

What occurs in the graph of

f

at the point

(2,3)

\newline

Choose

1

answer:

\newline

(A)

(2,3)

is a minimum point.

\newline

(B)

(2,3)

is a maximum point.

\newline

Given Information: We are given that $f$ is a twice differentiable function, which means it is smooth and has a continuous first and second derivative. We are also given that $f(2) = 3$ , $f'(2) = 0$ , and $f''(2) = 5$ . The value $f(2) = 3$ tells us that the point $(2,3)$ lies on the graph of $f$ . The value $f'(2) = 0$ indicates that the slope of the tangent to the graph of $f$ at $x = 2$ is zero, which means the graph has a horizontal tangent line at this point. This could be indicative of a local maximum, local minimum, or a point of inflection.
Analysis of $f'(2)$ : To determine whether $(2,3)$ is a maximum, minimum, or neither, we need to consider the second derivative, $f''(2)$ . The value $f''(2) = 5$ is positive, which tells us that the concavity of the graph at $x = 2$ is upwards (like a smile). According to the second derivative test, if the first derivative at a point is zero and the second derivative at that point is positive, then the function has a local minimum at that point.
Second Derivative Test: Since $f'(2) = 0$ and f''(2) > 0, we can conclude that $(2,3)$ is a local minimum point on the graph of $f$ . This answers the question prompt.