Final answer:
The function f(x) = (x+2)^5 (x-3)^4 has two critical points, which are found by setting its derivative equal to zero and solving for x.
Step-by-step explanation:
The number of critical points of a function is found by taking the derivative of the function and setting it equal to zero to solve for values of x. The function in question is f(x) = (x+2)^5 (x-3)^4. To find the critical points, we differentiate the function:
f'(x) = 5(x+2)⁴(x-3)⁴ + 4(x+2)⁵(x-3)³
To find the critical points, we set the derivative equal to zero:
0 = f'(x) = 5(x+2)⁴(x-3)⁴ + 4(x+2)⁵(x-3)³
The x values that make this true are x = -2 and x = 3, as these would make a factor in the derivative equal to zero. This means the function f(x) has two critical points.