Below is the graph of the derivative f\'(x) of a function defined on the interva

Question

Below is the graph of the derivative f'(x) of a function defined on the interval (0,8). You can click on the graph to see a larger version in a separate window. Refer to the graph to answer each of the following questions. For parts (A) and (B), use interval notation to report your answer. (If needed, you use U for the union symbol.) For what values of x in (0,8) is f(x) increasing? (If the function is not increasing anywhere, enter None .) For what values of A: in (0,8) is f(x) concave down? (If the function is not concave down anywhere, enter None .) Find all values of a: in (0,8) is where f(x) has a local minimum, and list them (separated by commas) in the box below. (If there are no local minima, enter None.) Find all values of x in (0,8) is where f(x) has an inflection point, and list them (separated by commas) in the box below. (If there are no inflection points, enter None.)

Explanation / Answer

let f be any differential function on interval (0,8)

then

if f'(x)<0,f(x) is decreasing

if f'(x)>0,f(x) is increasing

if f'(x) = 0,f(x) is constant

1. f'(x) is greater than 0 from 3 to 8

so f(x) is increasing at (3,8)

2. NONE

a graph f(x) is concave down when first function is increasing and than constant and then decreasing.

so first f'(x) should be >0,then equal to 0 and after that less than 0.

but here the sequence is opposite.so the f(x) is concave up whole time and not concave down at any point.

3.(3)

for a local minima to exist f'(x) should be equal to 0 and its lhs should be less than 0. and rhs should be greater than 0.

so local minima is at (3) only.

4. NONE

as for inflections to have concavity should change but the whole function here is concave up.

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