Final answer:
To find the value(s) of x where the function has a point of inflection, we need to find where the concavity changes. By finding the second derivative of the function and setting it equal to zero, we can solve for x and determine the points of inflection.
Step-by-step explanation:
To determine the value(s) of x for which the function f(x) = x⁴ + 10x³ + 36x² + 12x + 2 has a point of inflection, we need to find where the concavity changes. The concavity of a function changes at points where the second derivative is equal to zero or undefined. So, we need to find the second derivative of f(x) and solve for x when it is equal to zero.
First, let's find the first derivative of f(x):
f'(x) = 4x³ + 30x² + 72x + 12
Then, let's find the second derivative by taking the derivative of f'(x):
f''(x) = 12x² + 60x + 72
Now, we can set the second derivative equal to zero and solve for x:
12x² + 60x + 72 = 0
This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula.
By using the quadratic formula, we find that x ≈ -6.46 and x ≈ -2.73. Thus, the function f(x) = x⁴ + 10x³ + 36x² + 12x + 2 has points of inflection at x ≈ -6.46 and x ≈ -2.73.