Question

Can someone solve this differentiation question.

Answer 1

We clearly see on the graph that :
A. f is increasing on (2,6) and (8,10) (the derivative is >=0)
B. f is decreasing on (0,2) and (6,8) (the derivative is <=0)
C. f has two relative minima : one at x=2 and one at x=8 (the derivative changes signs there from negative to positive)
D. f has two relative maxima : one at x=6 and one at x=10(the derivative changes signs there from positive to negative)
E. f is concave up when f' is increasing i.e. on (0,4) and (7,9)
F. f is concave down when f' is decreasing i.e. on (4,7) and (9,10)
G. the points of inflexion of f are the points at which f' has an horizontal tangent, thus they are at x=4, x=7 and x=9
H. see the picture attached

Answer 2

A).  The function is increasing where its derivative is positive.
Its derivative is positive from 2 to 6, and from 8 to 10.

B).  The function is decreasing where its derivative is negative.
Its derivative is negative from 0 to 2, and from 6 to 8.

C).  The function has a relative minimum where its derivative is zero
and changing from negative to positive.
Its derivative is zero and changing from negative to positive at 2 and 8.

D).  The function has a relative maximum where its derivative is zero
and changing from positive to negative.
Its derivative is zero and changing from positive to negative at 6 and 10.

E).  The function is concave up between consecutive relative maxima.
The interval between consecutive relative maxima is  6 < x < 10 .

F).  The function is concave down between consecutive relative minima.
The interval between consecutive relative minima is  2< x < 8 .

G).  The function has points of inflection where its second derivative is
zero, that is, where its first derivative is a relative minimum or a relative
maximum.
Its first derivative is a relative minimum or maximum at x =  0,  4,  7, and  9 .

H).  Good luck on the sketch !