Question

Find the minimum value of (f(x) = -3/4 x^3 + 3/4x^4 when x belongs to the real numbers.

A) -3/4
B) 0
C) 3/4
D) 1

Answer 1

To find the minimum value of the function f(x) = -3/4x^3 + 3/4x^4, we take its derivative and find its critical points. Evaluating the function at these points, we determine that the minimum value is 3/4.

Step-by-step explanation:

To find the minimum value of the function f(x) = -3/4x^3 + 3/4x^4, we need to find its critical points. We can do this by taking the derivative of the function and setting it equal to zero. Taking the derivative of f(x), we get f'(x) = -9/4x^2 + 12/4x^3. Setting this equal to zero, we have -9/4x^2 + 12/4x^3 = 0. Simplifying, we get -9x^2 + 12x^3 = 0. Factoring out an x^2, we have x^2(-9 + 12x) = 0. This equation has two solutions: x = 0 and x = 9/12 = 3/4.

To determine which one of these values gives the minimum, we can evaluate the function at each point. Plugging in x = 0, we get f(0) = 0. Plugging in x = 3/4, we get f(3/4) = -3/4(3/4)^3 + 3/4(3/4)^4 = 3/4.

Therefore, the minimum value of f(x) is 3/4, so the correct answer is C) 3/4.