The accompanying figure shows the graph of the derivative of a function h that is defined and continuous on the interval − ∞ , + ∞ . Assume that the graph of h ′ has a vertical asymptote at x = 3 and that h ′ x → 0 + as x → − ∞ h ′ x → − ∞ as x → + ∞ (a) What are the critical points for h x ? (b) Identify the intervals on which h x is increasing. (c) Identify the x -coordinates of relative extrema for h x and classify each as a relative maximum or relative minimum. (d) Estimate the x -coordinates of inflection points for h x .
The accompanying figure shows the graph of the derivative of a function h that is defined and continuous on the interval − ∞ , + ∞ . Assume that the graph of h ′ has a vertical asymptote at x = 3 and that h ′ x → 0 + as x → − ∞ h ′ x → − ∞ as x → + ∞ (a) What are the critical points for h x ? (b) Identify the intervals on which h x is increasing. (c) Identify the x -coordinates of relative extrema for h x and classify each as a relative maximum or relative minimum. (d) Estimate the x -coordinates of inflection points for h x .
The accompanying figure shows the graph of the derivative of a function
h
that is defined and continuous on the interval
−
∞
,
+
∞
. Assume that the graph of
h
′
has a vertical asymptote at
x
=
3
and that
h
′
x
→
0
+
as
x
→
−
∞
h
′
x
→
−
∞
as
x
→
+
∞
(a) What are the critical points for
h
x
?
(b) Identify the intervals on which
h
x
is increasing.
(c) Identify the x-coordinates of relative extrema for
h
x
and classify each as a relative maximum or relative minimum.
(d) Estimate the x-coordinates of inflection points for
h
x
.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
Suppose that f:R → R is a function with second derivative f"(x)=x•(x+ 1)³ • (x + 5)ª .
Find intervals of concavity-up and concavity-down of f(x), and the x-coordinate(s) of any point(s) of inflection. Explain your
answers.
3. The following graph represents the derivative function f'(x) of a function f(x).
a) Find the interval of increase and decrease of the function
b) Find the intervals of concavity
/4A
-3 -2
Prob. 6 (a) (10 point) Let f(x) = 2x² – 3. Find ƒ'(−2) using only the limit definition of
derivatives.
(b) (10 p.) If ƒ(x) = √√x + 6, find the derivative f'(c) at an arbitrary point c using only the
limit definition of derivatives.
University Calculus: Early Transcendentals (4th Edition)
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