Please HELP with this quick question?

Question

Let f(x) = (2x – 5)^5. When is f(x) concave down?

a) x > 5/2
b) x < 5/2
c) all x
d) never

Answer 1

f' = 5(2x - 5)^4 (2) = 10(2x - 5)^4
f'' = 4(10)(2x - 5)^3 (2) = 80 (2x - 5)^3

find x when f'' = 0

80(2x - 5)^3 = 0
2x - 5 = 0
x = 5/2

Check a point on either side of that x value...
x = 1
f'' = 80(2 - 5)^3 = negative

x = 3
f'' = 80(6 - 5)^3 = positive

f(x) is concave UP when x > 5/2
and concave DOWN when x < 5/2

Answer 2

f(x) is an odd function (to the fifth power), so c and d are both impossible. The question is what is f(x) doing on either side of 5/2

f(1) = (2-5)^5 = (-3)^5 = -243
f(2) = (4-5)^5 = (-1)^5 = -1
f(3) = (6-5)^5 = 1^5 = 1
f(4) = (8-5)^5 = 3^5 = 243

So the answer is b. f(x) is concave down when x<5/2

Answer 3

Ok, to analyze that, we need to find the second derivative :

F(x) = (2x - 5)^5

F'(x) = 5*(2x - 5)^4*2

F'(x) = 10*(2x - 5)^4

F''(x) = 40*(2x - 5)^3*2

F''(x) = 80*(2x - 5)^3 >>>>> Now, let's analyze this

The function is concave down, when : F''(x) < 0

80*(2x - 5)^3 < 0

(2x - 5)^3 < 0

x < 5 / 2

Answer will be b)

The function is concave down, for x < 5 / 2.

That's it.