Answer :
Final Answer:
The critical numbers of the function[tex]\(f(x) = 3x^4 + 8x^3 - 48x^2\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Explanation:
To find the critical numbers, we need to locate the values of [tex]\(x\)[/tex] where the derivative of the function is equal to zero or undefined. The derivative [tex]\(f'(x)\)[/tex]can be determined by taking the derivative of each term in the original function.
[tex]\[f'(x) = 12x^3 + 24x^2 - 96x\][/tex]
Now, set[tex]\(f'(x) = 0\)[/tex]and solve for [tex]\(x\)[/tex]:
[tex]\[12x^3 + 24x^2 - 96x = 0\][/tex]
Factoring out [tex]\(12x\)[/tex], we get:
[tex]\[12x(x^2 + 2x - 8) = 0\][/tex]
This equation has solutions [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex]. These are the critical numbers.
The first derivative test can be employed to determine whether these critical points correspond to relative maxima, minima, or points of inflection. However, for the purpose of identifying critical numbers, we can conclude that [tex]\(x = 0\)[/tex]and [tex]\(x = 2\)[/tex] are the points where [tex]\(f(x)\)[/tex] may have a turning point or an extremum.
In summary, the critical numbers for the given function[tex]\(f(x) = 3x^4 + 8x^3 - 48x^2\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
