# 3.3.291 AP Calculus Exam Problem solve for k

• MHB
• karush
In summary, to find the value of $k$ for which the function $g(x)$ has a critical point at $x=\dfrac{2}{3}$, we can plug in the given value for $x$ and set the derivative of $g(x)$ equal to zero. Solving for $k$, we get $k=-\dfrac{3}{2}$, which corresponds to option (C).
karush
Gold Member
MHB
Let $g(x)$ be the function given by $g(x) = x^2e^{kx}$ , where k is a constant. For what value of k does g have a critical point at $x=\dfrac{2}{3}$?
$$(A)\quad {-3} \quad (B)\quad -\dfrac{3}{2} \quad (C)\quad -\dfrac{3}{2} \quad (D)\quad {0} \quad (E)\text{ There is no such k}$$

ok I really did not know how you could isolate k to solve this.

if you plug in $x=\dfrac{2}{3}$ then g(2/3)

if you plug in $x=\dfrac{2}{3}$ then $g(2/3)=\dfrac{4}{9}e^\dfrac{2k}{3}$

Last edited:
a critical point is a value(s) of $x$ where the derivative equals zero or is undefined

$g(x) = x^2 \cdot e^{kx}$

$g'(x) = x^2 \cdot ke^{kx} + 2x \cdot e^{kx}$

$x^2 \cdot ke^{kx} + 2x \cdot e^{kx} = 0 \implies xe^{kx}(kx + 2) = 0$

at $x = \dfrac{2}{3}$, the first factor is always > 0

the second factor ...

$\left(k \cdot \dfrac{2}{3} + 2 \right) = 0 \implies k = ?$

## What is the "3.3.291 AP Calculus Exam Problem solve for k"?

The "3.3.291 AP Calculus Exam Problem solve for k" refers to a specific problem on the AP Calculus Exam that asks students to solve for the value of k in a given equation or scenario.

## Why is solving for k important in calculus?

Solving for k is important in calculus because it allows us to find the specific value or constant that satisfies a given equation or problem. This helps us better understand the behavior and relationships between different variables in a function.

## What are the steps to solve for k in a calculus problem?

The steps to solve for k in a calculus problem may vary depending on the specific problem, but generally, it involves isolating the variable k on one side of the equation and using algebraic manipulation to solve for its value.

## What are some common strategies for solving for k in calculus?

Some common strategies for solving for k in calculus include using the properties of logarithms, substitution, and solving systems of equations. It is also important to carefully analyze the given problem and use critical thinking skills to determine the most efficient approach.

## Are there any tips for successfully solving for k in calculus?

Some tips for successfully solving for k in calculus include practicing regularly, understanding the basic principles and rules of calculus, and seeking help from a teacher or tutor if needed. It is also important to carefully read and understand the problem before attempting to solve it.

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