**(d) Determine the inflection points.**In mathematical terms, an inflection point is a point on a curve at which the curve changes concavity, which means it goes from being concave (curving upward) to convex (curving downward), or vice versa. To find the inflection points of a function, one typically needs to find where the second derivative of the function changes sign. **Example Calculation:**1. Given a function \( f(x) \), first, find its first derivative \( f'(x) \).2. Next, find the second derivative \( f''(x) \).3. Solve the equation \( f''(x) = 0 \) to find potential inflection points.4. Verify if the sign of \( f''(x) \) changes around those points.**Graph Explanation:**If a graph were provided, we would look for points where the curve changes its direction of curvature. These points indicate where the function switches from a "bowl-shaped" upward curve to a "cap-shaped" downward curve (or vice versa), confirming the presence of an inflection point. ### Function Analysis and DomainConsider the function \( f(x) = e^{-x} \cos(x) \), with the domain \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).**Instructions:**- Provide answers in exact form.### Explanation:- **Function**: The function \( f(x) \) is defined as the product of an exponential function and a trigonometric function: \[ f(x) = e^{-x} \cos(x) \]- **Domain**: The function is considered in the interval: \[ -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \]- **Guideline**: Ensure that all answers derived from the function and within its domain should be expressed in exact form without numerical approximations.This function combines exponential decay and oscillatory behavior, offering a complex example to study within the specified interval. The exponential part \(e^{-x}\) ensures that the function decreases as \(x\) increases, whereas the cosine part \(\cos(x)\) introduces cyclical variations. The interplay between these two types of functions can be explored in various mathematical contexts.

Name: **(d) Determine the inflection points.**In mathematical terms, an inf
Uploaded: 2024-03-20T06:46:42.000Z
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Description: **(d) Determine the inflection points.**In mathematical terms, an inflection point is a point on a curve at which the curve changes concavity, which means it goes from being concave (curving upward) to convex (curving downward), or vice versa. To fi