The given problem involves finding the values of $a$ and $b$ for the function $f(x) = 3x^5 - 30x^3 + ax^2 + bx + 60x + 1$ under the condition that the function has inflection points at $x = 3$ and $x = 2$.

To solve this, follow these steps:

1. Inflection Point Condition

Inflection points occur where the second derivative of a function changes sign, which means the second derivative equals zero at those points.

2. First Derivative

Calculate the first derivative of $f(x)$: $f'(x) = 15x^4 - 90x^2 + 2ax + b$

3. Second Derivative

Take the second derivative: $f''(x) = 60x^3 - 180x + 2a$

4. Solve for $a$ and $b$ Using Inflection Points

At $x = 3$ and $x = 2$, the second derivative must equal zero.

Step 4.1: Substitute $x = 3$:

$f''(3) = 60(3)^3 - 180(3) + 2a = 0$ $f''(3) = 60(27) - 540 + 2a = 0$ $1620 - 540 + 2a = 0$ $2a = -1080 \quad \Rightarrow \quad a = -540$

Step 4.2: Substitute $x = 2$:

$f''(2) = 60(2)^3 - 180(2) + 2a = 0$ $f''(2) = 60(8) - 360 + 2(-540) = 0$ $480 - 360 - 1080 = 0$ $-960 = 0 \quad \text{(inconsistent, likely typo in problem)}$

Let me check again for any error and re-solve with better clarity. sympy import symbols, Eq, solve

Define variables

x, a, b = symbols('x a b')

Function and derivatives

f_double_prime = 60x**3 - 180x + 2*a

Conditions for inflection points at x=3 and x=2

eq1 = Eq(f_double_prime.subs(x, 3), 0) eq2 = Eq(f_double_prime.subs(x, 2), 0)

Solve for a

solutions = solve((eq1, eq2), (a, b)) solutionsIt seems there might be an inconsistency or missing information in the problem, as no valid solutions for $a$ and $b$ satisfy the given conditions of inflection points at both $x = 3$ and $x = 2$. Could you confirm if the problem statement is correct or provide additional details?