3.1-3.5 Alternate Assessment
Absolute Extrema : - The absolute highest and lowest points of a graph.- Can only occur at endpoints or critical values.
Extreme Value Theorem (EVT) : If a function f is continuous on the closed interval [a,b], then it has both an absolute maximum and an absolute minimum value on the interval [a,b].
What are the critical values of the cube root of (x^2-x)? : x = 1/2, x = 0, x = 1
Show that 5 is a critical value of the function g(x)=2+(x-5)^3, but g does not have local extrema at 5. : If you set g'(x)=0, you will find that a critical value equals 5. If you place 5 on a number line and test the intervals, however, it is positive on both sides. 5 is therefore not a local extrema.
Which of the following functions satisfy the hypothesis of the MVT? I. f(x)= 1/(x+1) on [0, 2]II. f(x)= x^(1/3) on [0, 1]III. f(x)= |x| on [-1, 1] : I and II
As a graduation present, Jenna received a sports car which she drives very fast but very, very smoothly and safely. She always covers the 53 miles from her apartment in Austin, Texas to her parents' home in New Braunfels in less than 48 minutes. To slow her down, her dad decides to change the speed limit (he has connections.) Which one of the speed limits below is the highest speed her father can post, but still catch her speeding at some point on her trip? : Jenna's average velocity is 66.25 mph, so her father should set the speed limit to 65 mph in order to catch her speeding.
Let f be a function defined on [−1,1] such that f(-1) = f(1). Consider the following properties that f might have: I. f is continuous on [−1,1], differentiable on (−1,1). II. f(x) = (cosx)^3 III. f(x) = |sinπx|Which properties ensure that there exists a c in (−1,1) at which f'(c) = 0? : I, II, and III
Determine if Rolle's Theorem can be applied to f(x) = cos2x on [(-π/12), (π/6)]. If so, find the value(s) guaranteed by the theorem. : Rolle's Theorem does not apply. Though the function is continuous and differentiable, f(-π/12) does not equal f(π/6).
When is f(x) = x[4+(x^2)-((x^4)/5)] increasing? : (-2, 2)
The derivative of a function f is given for all x by f'(x) = (2x^2+4x-16)(1+g^2(x)) where g is some unspecified function. At which value(s) of x will f have a local maximum? : Because g is squared, the other factor determines the sign. F has a local max at x=-4.
Relationship between f'' and concavity : A function is concave up when f''(x) > 0 and concave down when f''(x) < 0. When f''(x)=0, then f has no concavity.
Relationship between f' and f'' : When f''(x)>0, the slope of f' is increasing. When f''(x)<0, the slope of f' is decreasing.
When is h(x) concave up if h'(x) = (x^2-2)/x? : H(x) is concave up (−∞, 0) U (0, ∞).
A cubic polynomial function f is defined by f(x) = 4x^3+ax^2+bx+k, where a, b, and k are constants. The function f has a local minimum at x = −1, and the graph of f has a point of inflection at x = −2 . Find the values of a and b. : a = 24b = 36
What is the value of k for which f(x)=x+(k/x) has a relative minimum at x=3? : k = 9
A function y = f(x) has the properties that f'(a) = 0 and f′′(a) = 0. Which one of the following statements must be true? (A) The graph of y = f(x) has a horizontal tangent at (a, f(a)). (B) (a, f(a)) is a point of inflection (C) (a, f(a)) is either a local maximum or a local minimum point. (D) f may be discontinuous at x = a : A. Choices B and C are not always true. Choice D would not apply; f(x) has to be continuous at x = a to be differentiable.
If f'(x) = [-12(x-1)]/x^(2/3), when is f(x) decreasing and concave down? : (1, ∞)
Given any given function y = f(x), how many of the following statements MUST be true? I. If f′′(a) < 0, then the graph of y = f(x) is concave up at x = a.II. If f′(a) does not exist, then x = a is not in the domain of y = f(x). III. If f′(a) = 0 and f′′(a) > 0, then f(a) is a local max. IV. If f′(a) = DNE and f′(x) changes from negative to positive at x = a, then f(a) is a local min. : None of these statements are true.f''(x) < 0 means the graph of f(x) is concave down.x = a could occur at a discontinuity, cusp, etc.f''(x) > 0 (concave up), so f(a) is a local min.f(a) could be DNE at a vertical asymptote.