Let f be a twice-differentiable function

Question

Let f be a twice-differentiable function with derivative given by f'(x) = 4x³ - 24x².
(A) Find the x-coordinate of any possible critical points of f. Show your work.
(B) Find the x-coordinate of any possible inflection points of f. Show your work.
(C) Use the Second Derivative Test to determine any relative extrema and inflection points. Justify your answers.
(D) If f has only one critical point on the interval [5, 8], what is true about the function f on the interval [5, 8]? Justify your answers.

Kunduz · Accepted Answer

Finding
-
are
f'(x) = 4x² – 24x²
f (x) = (2x 48 xviv 4 ) 341- 4
Part A
the critical pt. f'(a) = 0
→ 43?- 24x30 > 4x2(2-6) = 0
=> x = 0,0,6
Now Se the co-ordinate
of
critical pt
086.!
Part B f(0) = 8 but f"(o)= 0 - 48= -48 to
So, x=0 is the pt. of inflection
36(12-8)= 36x4
= 1447 0
So, atx=0 f(x) has it's minimum value.
Clearly
x=6 € [5,8] and flag is having
()
minimum value at x=6.
Also f(x) 70 for a 7, 6 f is increasing in [6.8]
[]
f'(a) so for x [6 fis decreasing in [5,6]
[
[
So, f(x) has absolute minimum value at x=6.
Part C f (6) = 12+36-486 =
Part D

Let f be a twice-differentiable function with derivative given by f'(x) = 4x³ - 24x². (A) Find the x-coordinate of any possible critical points of f. Show your work. (B) Find the x-coordinate of any p