Final answer:
To find the relative extrema and horizontal point of inflection of the function f(x) = 3x^5 - 5x^3 + 1, we need to find the critical points and determine their nature. The critical points are x = 0, x = 1, and x = -1, which correspond to a relative maximum, a relative minimum, and horizontal points of inflection, respectively.
Step-by-step explanation:
To find the relative extrema and horizontal point of inflection of the function f(x) = 3x^5 - 5x^3 + 1, we need to find the critical points and determine their nature. Critical points occur where the derivative of the function is equal to zero or undefined. Let's start by finding the derivative:
f'(x) = 15x^4 - 15x^2
Setting f'(x) = 0, we can solve for x:
15x^4 - 15x^2 = 0
15x^2(x^2 - 1) = 0
x = 0, x = 1, x = -1
Now, we need to determine the nature of these critical points. We can analyze the sign of the second derivative f''(x) = 60x^3 - 30x:
f''(0) = 0, f''(1) = 30, f''(-1) = -30
At x = 0, the second derivative test is inconclusive. At x = 1, f''(x) > 0, so it is a relative minimum. At x = -1, f''(x) < 0, so it is a relative maximum. To find the horizontal point(s) of inflection, we need to set the second derivative equal to zero and solve:
60x^3 - 30x = 0
30x(x^2 - 1) = 0
x = 0, x = 1, x = -1
The points x = 0, x = 1, and x = -1 are the horizontal points of inflection.