Final answer:
To find f(4) and f(√4) for the function f(x) = 2 x³ - 9 x² + 5, plug in the values 4 and √4 respectively into the function. The critical numbers are obtained by deriving f, setting it to zero, and solving for x, which gives us the critical numbers 0 and 3.
Step-by-step explanation:
The student asks about specific evaluations and properties of the function f(x) = 2 x³ - 9 x² + 5. First, the student wants to find f(4) and f(√4) which are function evaluations for x=4 and x=√4, respectively. We calculate these by substituting the given values for x into the function. Secondly, the student is seeking the critical numbers of f, which are found by setting the derivative of f equal to zero and solving for x.
To find f(4), we calculate:
f(4) = 2(4)^3 - 9(4)^2 + 5 = 2(64) - 9(16) + 5 = 128 - 144 + 5 = -16 + 5 = -11.
To find f(√4), we calculate:
f(√4) = 2(√4)^3 - 9(√4)^2 + 5 = 2(2³) - 9(2^2) + 5 = 16 - 36 + 5 = -20 + 5 = -15.
To find the critical numbers, we first find the derivative of f:
f'(x) = 6x^2 - 18x
Then, set the derivative equal to zero to find the critical values:
0 = 6x^2 - 18x
x(6x - 18) = 0
x = 0 or 6x - 18 = 0
x = 0 or x = 3
Thus, the critical numbers are 0 and 3.