Let g(x)=2^(-x^(3)+5x). Below is Lila's attempt to write a formal justification for the fact that the equation g^(')(x)=-12 has a solution in the interval (1,2). Is Lila's justification complete? If not, why? Lila's justification: Exponential and polynomial functions are differentiable and continuous at all points in their domain, and [1,2] is within g^(')s domain. So, according to the mean value theorem, g^(')(x)=-12 must have a solution somewhere between x=1 and x=2. Choose 1 answer: (A) Yes, Lila's justification is complete. (B) No, Lila didn't establish that the average rate of change of g over [1,2] is equal to -12 . (C) No, Lila didn't establish that g is differentiable.

Mean Value Theorem Application: Lila's justification relies on the Mean Value Theorem, which states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists some c in (a, b) such that the function's derivative at c is equal to the average rate of change of the function over [a, b].Function Continuity and Differentiability: To apply the Mean Value Theorem, Lila must first ensure that g(x) is continuous on [1, 2] and differentiable on (1, 2). Since g(x) is an exponential function composed with a polynomial, it is continuous and differentiable everywhere in its domain, which includes [1, 2].Calculate Average Rate of Change: Next, Lila must calculate the average rate of change of g(x) over the interval [1, 2]. This is done by evaluating g(2) - g(1) and dividing by 2 - 1. However, Lila has not provided these calculations or shown that the average rate of change is equal to -12.Incomplete Justification: Without establishing that the average rate of change of g(x) over [1, 2] is -12, Lila cannot conclude that g^(')(x) = -12 for some x in (1, 2) based on the Mean Value Theorem. Therefore, Lila's justification is incomplete.

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Let
g(x)=2^(-x^(3)+5x).
Below is Lila's attempt to write a formal justification for the fact that the equation
g^(')(x)=-12 has a solution in the interval
(1,2).
Is Lila's justification complete? If not, why?
Lila's justification:
Exponential and polynomial functions are differentiable and continuous at all points in their domain, and
[1,2] is within
g^(')s domain.
So, according to the mean value theorem,
g^(')(x)=-12 must have a solution somewhere between
x=1 and
x=2.
Choose 1 answer:
(A) Yes, Lila's justification is complete.
(B) No, Lila didn't establish that the average rate of change of
g over
[1,2] is equal to -12 .
(C) No, Lila didn't establish that
g is differentiable.

Let $g(x)=2^{-x^{3}+5 x}$ . $\newline$ Below is Lila's attempt to write a formal justification for the fact that the equation $g^{\prime}(x)=-12$ has a solution in the interval $(1,2)$ . $\newline$ Is Lila's justification complete? If not, why? $\newline$ Lila's justification: $\newline$ Exponential and polynomial functions are differentiable and continuous at all points in their domain, and $[1,2]$ is within $g^{\prime} s$ domain. $\newline$ So, according to the mean value theorem, $g^{\prime}(x)=-12$ must have a solution somewhere between $x=1$ and $x=2$ . $\newline$ Choose $1$ answer: $\newline$ (A) Yes, Lila's justification is complete. $\newline$ (B) No, Lila didn't establish that the average rate of change of $g$ over $[1,2]$ is equal to $-12$ . $\newline$ (C) No, Lila didn't establish that $g$ is differentiable.

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Grade 7

One-step inequalities: word problems

Full solution

Q. Let

g(x)=2^{-x^{3}+5 x}

\newline

Below is Lila's attempt to write a formal justification for the fact that the equation

g^{\prime}(x)=-12

has a solution in the interval

(1,2)

\newline

Is Lila's justification complete? If not, why?

\newline

Lila's justification:

\newline

Exponential and polynomial functions are differentiable and continuous at all points in their domain, and

[1,2]

is within

g^{\prime} s

domain.

\newline

So, according to the mean value theorem,

g^{\prime}(x)=-12

must have a solution somewhere between

x=1

and

x=2

\newline

Choose

1

answer:

\newline

(A) Yes, Lila's justification is complete.

\newline

(B) No, Lila didn't establish that the average rate of change of

g

over

[1,2]

is equal to

-12

\newline

g

is differentiable.

Mean Value Theorem Application: Lila's justification relies on the Mean Value Theorem, which states that if a function is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ , then there exists some $c$ in $(a, b)$ such that the function's derivative at $c$ is equal to the average rate of change of the function over $[a, b]$ .
Function Continuity and Differentiability: To apply the Mean Value Theorem, Lila must first ensure that $g(x)$ is continuous on $[1, 2]$ and differentiable on $(1, 2)$ . Since $g(x)$ is an exponential function composed with a polynomial, it is continuous and differentiable everywhere in its domain, which includes $[1, 2]$ .
Calculate Average Rate of Change: Next, Lila must calculate the average rate of change of $g(x)$ over the interval $[1, 2]$ . This is done by evaluating $g(2) - g(1)$ and dividing by $2 - 1$ . However, Lila has not provided these calculations or shown that the average rate of change is equal to $-12$ .
Incomplete Justification: Without establishing that the average rate of change of $g(x)$ over $[1, 2]$ is $-12$ , Lila cannot conclude that $g'(x) = -12$ for some $x$ in $(1, 2)$ based on the Mean Value Theorem. Therefore, Lila's justification is incomplete.