Final answer:
To find the point of inflection for the function y = 3x⁵ - 10x⁴, we calculate the second derivative and determine the value of x where it equals zero and indicates a change in concavity. The correct answer is x at 2, option c), since that is where the second derivative is zero and the concavity changes.
Step-by-step explanation:
To determine the points of inflection for the graph of the function y = 3x⁵ - 10x⁴, we need to find the values of x where the second derivative of the function changes sign. A point of inflection occurs where the second derivative is zero or undefined, and where there's also a change in concavity (from concave up to concave down, or vice versa).
First, we find the first derivative of y with respect to x:
- y' = d/dx(3x⁵ - 10x⁴) = 15x⁴ - 40x³
Next, we find the second derivative of y with respect to x:
- y'' = d/dx(15x⁴ - 40x³) = 60x³ - 120x²
To find the points of inflection, we set the second derivative equal to zero:
- 0 = 60x³ - 120x²
- x²(60x - 120) = 0
- x²(60(x - 2)) = 0
From this, we get two potential points of inflection at x = 0 or x = 2. To verify if these points are indeed points of inflection, we would check the sign of the second derivative before and after these values. Since x = 2 provides an actual change in concavity, this is a point of inflection. Option a) 'x' at ⁰ is not correct because there is no change in concavity at this point. Option b) 'x' at ¹ is not part of the solutions we found for the second derivative. Option c) 'x' at ² is correct because it is a solution to the second derivative being zero and there is a change in concavity here. Option d) 'x' at ³ is incorrect because the second derivative does not equal zero at this value.
The correct option is c) 'x' at ².