Question

Find all values of x for which f(x)=3x^4+4x^3−120x^2+120 has local maxima and minima, as well as the inflection points. Show all your work.

Answer 1

Explanation:

To find the local maxima and minima, we need to find the points where the first derivative of the function is equal to zero. To find the inflection points, we need to find the points where the second derivative is equal to zero.

The first derivative of the function is given by:

f'(x) = 12x^3 + 12x^2 - 240x

To find the values of x for which the first derivative is equal to zero, we can set f'(x) equal to zero and solve for x:

12x^3 + 12x^2 - 240x = 0

This equation can be factored as:

(4x - 10)(3x^2 + 2x) = 0

This equation has three solutions: x = 10/4, x = 0, and x = -2/3.

To find the inflection points, we need to find the values of x for which the second derivative is equal to zero. The second derivative is given by:

f''(x) = 36x^2 + 24x - 240

To find the values of x for which the second derivative is equal to zero, we can set f''(x) equal to zero and solve for x:

36x^2 + 24x - 240 = 0

This equation can be factored as:

(6x + 40)(6x - 6) = 0

This equation has two solutions: x = -40/6 and x = 6/6.

Therefore, the function has local maxima and minima at x = 10/4 and x = -2/3, and inflection points at x = -40/6 and x = 6/6.

Answer 2

the values of x for which f(x) = 3x^4+4x^3−120x^2+120 has local maxima and minima, as well as the inflection points, are x = 0

Explanation:

To find the values of x for which f(x) = 3x^4+4x^3−120x^2+120 has local maxima and minima, as well as the inflection points, you can start by finding the critical points of the function, which are the points where the derivative of the function is equal to 0 or is undefined.

To find the derivative of the function, you can use the power rule:

f'(x) = 4 * 3x^3 + 3 * 4x^2 - 2 * 120x

= 12x^3 + 12x^2 - 240x

To find the critical points of the function, you can set the derivative equal to 0 and solve:

12x^3 + 12x^2 - 240x = 0

x(12x^2 + 12x - 240) = 0

The solutions to this equation are x = 0, x = -10, and x = 8/3.

To find the inflection points of the function, you can find the second derivative of the function:

f''(x) = 36x^2 + 24x - 240

To find the inflection points, you can set the second derivative equal to 0 and solve:

36x^2 + 24x - 240 = 0

(6x - 20)(6x + 12) = 0

x = 10/3, x = -2