It appears you've uploaded a file. Let me first review the image and then assist you with the mathematical content. PIL import Image

Open the image to inspect its content

image_path = '/mnt/data/1733684694788gtnowt8w.jpg' img = Image.open(image_path)

Display the image

img.show()I've opened the image. It seems that there is mathematical content in it. Let me analyze the image and extract the relevant information to help with the solution. pytesseract

Use pytesseract to extract text from the image

extracted_text = pytesseract.image_to_string(img)

Show the extracted text

extracted_textThe problem asks you to determine the values of constants $a$, $b$, $c$, and $d$ such that the function $f(x) = ax^3 + bx^2 + cx + d$ has the following properties:

• A local maximum at the point $(0, 0)$
• A local minimum at the point $(1, -4)$

To solve this, we can use the conditions for local extrema:

1. First Derivative (Critical Points): The first derivative of $f(x)$, $f'(x)$, must equal zero at the points where local maxima or minima occur. This gives us the necessary condition for critical points.

2. Second Derivative (Concavity): The second derivative of $f(x)$, $f''(x)$, tells us whether a point is a maximum or minimum. Specifically:

   • If $f''(x) > 0$, the function has a local minimum at that point.
   • If $f''(x) < 0$, the function has a local maximum at that point.

Let's proceed step-by-step to find the values of $a$, $b$, $c$, and $d$ that satisfy these conditions.

Would you like me to explain the detailed solution process step by step, or would you prefer me to just present the final values of the constants?