Given
y = 4x3 − 48x2 + 5.
(a) Find the critical values of the function. (Enter your answers as a comma-separated list
To find the critical values of the function, we need to find the points where the derivative of the function is zero or undefined.
Step 1: Find the derivative of the function.
The derivative of y = 4x^3 − 48x^2 + 5 is found by taking the derivative of each term separately. The derivative of x^3 is 3x^2, the derivative of x^2 is 2x, and the derivative of a constant (5) is zero.
So, the derivative of y with respect to x is:
dy/dx = 3(4x^2) - 48(2x)
dy/dx = 12x^2 - 96x
Step 2: Set the derivative equal to zero and solve for x.
12x^2 - 96x = 0
Step 3: Factor out common terms.
12x(x - 8) = 0
Step 4: Set each factor equal to zero and solve for x.
12x = 0 or x - 8 = 0
Solving the first equation, we get:
x = 0
Solving the second equation, we get:
x = 8
So, the critical values of the function are x = 0 and x = 8.
Therefore, the comma-separated list of critical values is 0, 8.