https://file.aiquickdraw.com/mathbot/file/17267849668109lnbfwwr.jpg

To find the stationary points of the function \( y = x^3 + 4x^2 - 5x + 2 \), we need to follow these steps:

### Step 1: Differentiate the function.
We first find the first derivative of \( y \) with respect to \( x \), which is necessary to locate the stationary points.

\[
\frac{dy}{dx} = 3x^2 + 8x - 5
\]

### Step 2: Set the first derivative equal to zero.
Stationary points occur when the first derivative equals zero, so we solve:

\[
3x^2 + 8x - 5 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 3 \), \( b = 8 \), and \( c = -5 \).

### Step 3: Apply the quadratic formula.
Substitute the values of \( a \), \( b \), and \( c \) into the formula:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(3)(-5)}}{2(3)}
\]

Simplifying:

\[
x = \frac{-8 \pm \sqrt{64 + 60}}{6}
\]
\[
x = \frac{-8 \pm \sqrt{124}}{6}
\]
\[
x = \frac{-8 \pm 11.14}{6}
\]

So the two solutions are:

\[
x_1 = \frac{-8 + 11.14}{6} = \frac{3.14}{6} = 0.52
\]
\[
x_2 = \frac{-8 - 11.14}{6} = \frac{-19.14}{6} = -3.19
\]

### Step 4: Conclusion
The stationary points occur at \( x = 0.52 \) and \( x = -3.19 \), which matches the given values in the image.

Would you like any further clarification or details? Here are some related questions to expand on this:

1. What are the types of stationary points (e.g., maximum, minimum, or saddle points)?
2. How do you find the second derivative to determine the nature of stationary points?
3. Could there be other points of interest in this function, such as inflection points?
4. How does the graph of this cubic function behave near the stationary points?
5. How do these stationary points help in sketching the curve of the function?

**Tip:** Always check the second derivative to classify the stationary points—whether they are maxima, minima, or points of inflection.

The stationary points of y = x^3 + 4x^2 - 5x + 2 are (0.52) and (-3.19). Type your answer to 2 decimal places.

Learn how to find the stationary points of the cubic function y = x^3 + 4x^2 - 5x + 2 by solving the quadratic equation derived from the first derivative. This problem explores calculus concepts such as differentiation and stationary points, suitable for students in Grades 10-12.

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