which function has a tangent that is perpendicular to the tangent of f(0) = x^4 + 3x^3 +10.

To find the function that has a tangent perpendicular to the tangent of the function \( f(x) = x^4 + 3x^3 + 10 \) at \( x = 0 \), we need to follow these steps:

### Step 1: Find the slope of the tangent line to \( f(x) \) at \( x = 0 \).
The slope of the tangent line to \( f(x) \) at any point is given by the derivative \( f'(x) \). So, let's differentiate the function:

\[
f(x) = x^4 + 3x^3 + 10
\]
\[
f'(x) = 4x^3 + 9x^2
\]

Now, evaluate \( f'(x) \) at \( x = 0 \):

\[
f'(0) = 4(0)^3 + 9(0)^2 = 0
\]

So, the slope of the tangent line to \( f(x) \) at \( x = 0 \) is 0.

### Step 2: Find the slope of the perpendicular tangent.
A tangent line perpendicular to another has a slope that is the negative reciprocal of the original slope. Since the slope of the tangent line to \( f(x) \) at \( x = 0 \) is 0, we need to find the slope of a line perpendicular to a horizontal line (with slope 0).

A line perpendicular to a horizontal line has an undefined slope, which corresponds to a vertical line. Therefore, the tangent line we are looking for is vertical.

### Step 3: Determine the function whose tangent is vertical.
A vertical tangent occurs when the derivative of a function approaches infinity or is undefined. An example of such a function is:

\[
g(x) = \sqrt{x}
\]

At \( x = 0 \), the derivative of \( g(x) \) is undefined, which means the tangent line is vertical. Thus, the function \( g(x) = \sqrt{x} \) has a vertical tangent at \( x = 0 \), making its tangent line perpendicular to the tangent of \( f(x) = x^4 + 3x^3 + 10 \) at \( x = 0 \).

### Conclusion:
The function \( g(x) = \sqrt{x} \) has a tangent that is perpendicular to the tangent of \( f(x) = x^4 + 3x^3 + 10 \) at \( x = 0 \).

Would you like more details on any of these steps or concepts?

Here are 5 related questions for further exploration:
1. How do you find the equation of a tangent line at a given point on a curve?
2. What is the geometric meaning of a vertical tangent line?
3. How does the concept of perpendicular lines apply to tangent lines in general?
4. Can you have perpendicular tangents if the slopes are both undefined at a point?
5. How would the perpendicularity change if the slope were non-zero?

**Tip:** The slope of a tangent line can often reveal key information about the behavior of the function at that point, including whether it's increasing, decreasing, or flat.

Which function has a tangent that is perpendicular to the tangent of f(0) = x^4 + 3x^3 + 10?

Explore how to determine a function whose tangent line is perpendicular to the tangent of f(x) = x^4 + 3x^3 + 10 at x = 0. Learn the concepts of derivatives, perpendicular slopes, and vertical tangents with a detailed solution. Suitable for students in Grades 11-12 or College Calculus.

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