Use the graph of f in the figure to identify the following (assume that f" (0) < 0 f" ( b ) > 0, and f" ( g ) > 0): (A) the intervals on which f' ( x ) <0 (B) the intervals on which f' ( x ) > 0 (C) the intervals on which f ( x ) is increasing (D) the intervals on which f ( x ) is decreasing (E) the .v coordinate(s) of the point(s) where f ( x ) has a local maximum (F) the x coordinate(s) of the point(s) where f ( x ) has a local minimum (G) the intervals on which f" ( x ) < 0 (H) the intervals on which f" ( x ) > 0 (I) the intervals on which the graph of f is concave upward (J) the intervals on which the graph of f is concave downward (K) the .v coordinate(s) of the inflection point(s) (L) the horizontal asymptote(s) (M) the vertical asymptote(s)
Use the graph of f in the figure to identify the following (assume that f" (0) < 0 f" ( b ) > 0, and f" ( g ) > 0): (A) the intervals on which f' ( x ) <0 (B) the intervals on which f' ( x ) > 0 (C) the intervals on which f ( x ) is increasing (D) the intervals on which f ( x ) is decreasing (E) the .v coordinate(s) of the point(s) where f ( x ) has a local maximum (F) the x coordinate(s) of the point(s) where f ( x ) has a local minimum (G) the intervals on which f" ( x ) < 0 (H) the intervals on which f" ( x ) > 0 (I) the intervals on which the graph of f is concave upward (J) the intervals on which the graph of f is concave downward (K) the .v coordinate(s) of the inflection point(s) (L) the horizontal asymptote(s) (M) the vertical asymptote(s)
Solution Summary: The author analyzes the function y=f(x), f's decreasing in a given interval, and the value of '0.
Use the graph of f in the figure to identify the following (assume that f"(0) < 0 f"(b) > 0, and f"(g) > 0):
(A) the intervals on which f'(x)<0
(B) the intervals on which f'(x) > 0
(C) the intervals on which f(x) is increasing
(D) the intervals on which f(x) is decreasing
(E) the .v coordinate(s) of the point(s) where f (x) has a local maximum
(F) the x coordinate(s) of the point(s) where f(x) has a local minimum
(G) the intervals on which f"(x) < 0
(H) the intervals on which f"(x) > 0
(I) the intervals on which the graph of f is concave upward
(J) the intervals on which the graph of f is concave downward
(K) the .v coordinate(s) of the inflection point(s)
(L) the horizontal asymptote(s)
(M) the vertical asymptote(s)
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 .
4.
5.
6.
Prove that p (gp) is a tautology using the laws of propositional logic.
Prove that p((pVq) → q) is a tautology using the laws of propositional logic.
Let us say a natural number n is ok if there are two natural numbers whose sum
is n and whose product is n. (Convention: the natural numbers consist of 0, 1, 2,...)
(a) Give a logical expression that means "n is ok".
(b) Show that 0 and 4 are both ok.
(c) Give a logical expression that means "every natural number is ok".
(d) Give a logical expression that means "it is not the case that every number is ok". Push
the negations into the expression as far as possible.
7.
Let E(x, y) be a two-variable predicate meaning "x likes to eat y", where the
domain of x is people and the domain of y is foods. Write logical expressions that represent
the following English propositions:
(a) Alice doesn't like to eat pizza.
(b) Everybody likes to eat at least one food.
(c) Every student likes to eat at least one food other than pizza.
(d) Everyone other than Alice likes to eat at least two different foods.
(e) There are two different people that like to eat the same food.
21. Determine for which values of m the function (x) = x™ is a solution to the given equation.
a. 3x2
d²y
dx²
b. x2 d²y
+11x
dy
- 3y = 0
dx
dy
dx2
x dx
5y
= 0
Chapter 4 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
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