label the relatvie extrema, point of inflection, asymptote and graph

Question

f(x)=4x^3-x^4

 

f(x)=(1)/(1+x^2)

Answer 1

The function is f(x) = 4x³ - x⁴.

f^,(x) = 12x² - 4x3.

f^,,(x) = 24x - 12x².

To find the critical or key numbers, to make the first derivative equal to zero or f ' (x) does not exist.

f^,(x) = 12x² - 4x3 = 4x² (3 - x) = 0.

The key numbers are x = 0 and x = 3.

(b).

TEST FOR INCREASING AND DECREASING FUNCTIONS :

If f^,(x) > 0 (Positive) for all x in (a, b), then f(x) is increasing on [a, b].

If f^,(x) < 0 (Negative) for all x in (a, b), then f(x) is decreasing on [a, b].

Test intervals    x - Value                    Sign of f^,(x)                             Conclusion

(- ∞, 0)              x = - 1         12(-1)² - 4(-1)3 = 12 + 4 = 16 > 0        Increasing

(0, 3)                 x = 1         12(1)² - 4(1)3 = 12 - 4 = 8 > 0               Increasing

(3, ∞)                x = 4          12(4)² - 4(4)3 = 192 - 256 = - 64 < 0    Decreasing

So, f(x) is increasing on the interval (- ∞, 3) and decreasing on the interval (3, ∞).

Find Relative Extrema :
The First Derivative Test :
Let f be a differential function with f(c) = 0 then

• If f ' (x) changes from positive to negative, f has a relative (local) maximum at c.
• If f ' (x) changes from negative to positive, f has a relative (local) minimum at c.

The function f(x) has relative maximum at x = 3, because f'(x) changes from positive to negative around 3.

f(3) = 4(3)³ - 3⁴ =108 - 81 = 27.

The relative maximum point (3, 27).

Points of Inflection : If (c, f(c)) is a point of inflection of the graph of f(x), then either f " (c) = 0 or f " (x) does not exist.

f^,,(x) = 24x - 12x² = 12x(2 - x) = 0.

The key numbers are x = 0 and x = 2.

If x = 0 then, f(0) = 4(0)³ - (0)⁴ = 0.

If x = 2 then, f(2) = 4(2)³ - (2)⁴ = 32 - 16 = 16.

The Inflection points are (0, 0) and (2, 16).

Graph :

Answer 2

(2).

The function is f(x) = 1/(1 + x²).

To find the vertical asymptotes, denominator function is zero.

1 + x² = 0

x² = - 1

x = ± i.

Since the imaginary numbers are negligible, there is no vertical asymptotes.

Since degree of numerator (0) < degree of denominator (2), the horizontal asymptote y = 0.

The function is .

Differentiate with respect to x.

Again differentiate with respect to x.

To find the critical numbers, or does not exist.

The function does not exist when .

The imaginary numbers are negligible, so the critical number is x = 0.

Apply second derivative test:

The relative maximum point (0, 1).

To find the points of inflection, or does not exist.

The function does not exist when 

The imaginary numbers are negligible.

The points of inflections are (1, 1/2) and (-1, 1/2).

Graph :