9. graph

Example 1: f(x) = x⁴ + 8x³ + 22x² + 24x + 9

•1. Domf = R

•2. f is continuous and derivable in R because it is a polynomial function.

•3. Symmetry: f is a function which is neither odd nor even.

•4. f is not a periodic function.

•5. intersection points: (-3,0),(-1,0),(0,9)

•6. There are no asymptotes because it is a polynomial function.

•7. f’(x) = 4x³ + 24x² + 44x + 24                 f’(x) = 0 ↔ x Є {-3, -2, -1}

Example 2:

•1. Domf = R – {-2,2}

•2. f is continuous and derivable in its domain because it is a rational function.

•3. Symmetry: f is an even function.

•4. f is not a periodic function.

•5. intersection points: (-3,0),(3,0),(0,9/4)

•6. Asymptotes: x = 2; x = -2; y = 1

•7.

•8.

no solution

 

 

•9. Graph

 

Example 3 (PAEG September 2014):

a) Study its continuity in x = -1

b) Find the relative extrema in the interval (1,4)

c) Calculate the intervals of increasing and decreasing in the interval (1,4)

 

a) f is discontinuous in x = -1, it has a jump discontinuity with jump 1

 

b) 

  It has a  minimum in x = 2

c) f is decreasing in the interval (1,2) and increasing in the interval (2,4)                    

 

Exercise: draw the graphs of the functions:

a) f(x) = x·e^x

 

 

 

 

 

 

 

 

Solutions:

a)

b)