At what point does the curve have maximum curvature?
y = 6e^x

To determine the point at which the curve \( y = 6e^x \) has maximum curvature, we need to compute the curvature formula and find its maximum value.

The curvature \( \kappa(x) \) of a curve given by \( y = f(x) \) is:

\[
\kappa(x) = \frac{|f''(x)|}{\left( 1 + (f'(x))^2 \right)^{3/2}}
\]

### Step 1: Find the first and second derivatives of \( y = 6e^x \).

1. The first derivative \( f'(x) \) is:
\[
f'(x) = \frac{d}{dx}(6e^x) = 6e^x
\]

2. The second derivative \( f''(x) \) is:
\[
f''(x) = \frac{d}{dx}(6e^x) = 6e^x
\]

### Step 2: Plug the derivatives into the curvature formula.
Now that we have \( f'(x) = 6e^x \) and \( f''(x) = 6e^x \), we substitute them into the curvature formula:

\[
\kappa(x) = \frac{|6e^x|}{\left( 1 + (6e^x)^2 \right)^{3/2}}
\]

This simplifies to:

\[
\kappa(x) = \frac{6e^x}{\left( 1 + 36e^{2x} \right)^{3/2}}
\]

### Step 3: Maximize the curvature.
To find the point where the curvature is maximized, we take the derivative of \( \kappa(x) \) with respect to \( x \), set it equal to zero, and solve for \( x \).

Let's compute this derivative.

\[
\kappa'(x) = \frac{d}{dx} \left( \frac{6e^x}{\left( 1 + 36e^{2x} \right)^{3/2}} \right)
\]

Using the quotient rule and simplifying might give us the critical points where the maximum occurs.

Let me now compute this derivative and find where it equals zero.The real critical point occurs at approximately \( x \approx -2.138 \). This is where the curve \( y = 6e^x \) has maximum curvature.

Would you like more details on how this result was derived, or have any other questions?

Here are 5 related questions you might find interesting:
1. What is the curvature of the curve at \( x = -2.138 \)?
2. How does curvature behave as \( x \to \infty \) for exponential functions like \( 6e^x \)?
3. Can the point of maximum curvature be determined without derivatives for some simpler functions?
4. How does curvature change for a polynomial function compared to an exponential one?
5. What physical interpretations can be linked to the concept of curvature?

**Tip:** Curvature is particularly useful in geometry and physics to understand how sharply a curve bends at a particular point!

At what point does the curve have maximum curvature? y = 6e^x

Find the point of maximum curvature for the curve y = 6e^x by using the curvature formula. Learn step-by-step how to differentiate and apply the curvature equation. This problem explores key calculus concepts like derivatives and curvature, suitable for students in Grades 11-12 or early college.

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