Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = 4x3 + x2 + 4x
well, f'(x) = 12x^2+2x+4 = 2(6x^2+x+2)
so, keeping in mind the definition of critical numbers, whet do you think?
you can't factor it and also can't quadratic 4X^3+X^2+4X so the answer is none(DNE).
Finding critical numbers involves taking the derivative of the function and setting it equal to zero. So, let's differentiate f(x) = 4x^3 + x^2 + 4x:
f'(x) = 12x^2 + 2x + 4
Next, let's find the values of x that make f'(x) equal to zero:
12x^2 + 2x + 4 = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-2 ± √(2^2 - 4(12)(4))) / (2)(12)
x = (-2 ± √(4 - 192)) / 24
x = (-2 ± √(-188)) / 24
Since we cannot take the square root of a negative number, the expression inside the square root is imaginary, which means there are no real solutions for x. Thus, the critical numbers for the function f(x) = 4x^3 + x^2 + 4x are DNE (does not exist).
Well, it seems like the function is one tough cookie! No critical numbers for this one. Keep searching, my friend!
To find the critical numbers of a function, we need to find the values of x where the derivative of the function is either zero or undefined.
1. Start by finding the derivative of the given function, f(x) = 4x^3 + x^2 + 4x. The derivative of a term with exponent n can be found by multiplying the coefficient by n and subtracting 1 from the exponent.
f'(x) = 12x^2 + 2x + 4
2. Set the derivative equal to zero and solve for x.
12x^2 + 2x + 4 = 0
Unfortunately, the quadratic equation 12x^2 + 2x + 4 = 0 does not factor, so we need to use the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 12, b = 2, and c = 4. Plugging in these values:
x = (-2 ± √(2^2 - 4*12*4)) / (2*12)
x = (-2 ± √(4 - 192)) / 24
x = (-2 ± √(-188)) / 24
The expression √(-188) is not real, which means there are no real values of x that make the derivative equal to zero. Therefore, there are no critical numbers for this function.
Answer: DNE (Does Not Exist)