Let f(x) be a strictly increasing function such that f(x)<0 and g(x) be a strictly decreasing function such that g(x)>0 for all values of x.  If $h\left(x\right)=f\left(x\right)g\left(x\right)$ , which statement is true? 

The graph of y=h(x) is shown. Which of the following could be the graph of y=h'(x)?

The figure shows the graph of f', the derivative of f, for [-7,7]. The graph of f' has horizontal tangents at x=−3, x=2, and x=5 and a vertical tangent at  x=3.

Find all values of x, for (-7,7), at which f attains a relative minimum. 

The figure shows the graph of f', the derivative of f, for [-7,7]. The graph of f' has horizontal tangents at x=−3, x=2, and x=5 and a vertical tangent at  x=3.

Find all values of x, for (-7,7), at which f attains a relative maximum. 

The figure shows the graph of f', the derivative of f, for [-7,7]. The graph of f' has horizontal tangents at

The figure shows the graph of f', the derivative of f, for [-7,7]. The graph of f' has horizontal tangents at

The graph of the second derivative of a function f(x) is shown. Select all of the following that are true.

Given the graph of the derivative, f'(x), which of the following would correctly depict a possible graph of f(x)?

Given the graph of f(x), how many x values satisfy the result of the Mean Value Theorem on the interval [0,7]?

Identify the interval(s) on which the function $f\left(x\right)=x^2-x-12$ is increasing. 

Suppose f(x) is differentiable everywhere and $f\left(-2\right)=-5$ and $f'\left(x\right)\le5$ for all values of x.  Using the mean value theorem, what is the largest possible value of $f\left(8\right)$ ? 

For all x in the closed interval [2,5] the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f?

On the closed interval  $$f\left(x\right)=e^{\sin x}$$  , the absolute minimum of $$\left[0,2\pi\right]$$ occurs at which x value?

Let g be the function $g\left(x\right)=x^2e^{kx}$ where k is a constant.  For what values of k does g have a critical point at  $x=\frac{2}{3}$  ? 

The maximum value of $f\left(x\right)=2x^3-15x^2+36x$  on the closed interval [0,4] is

Consider the function $f\left(x\right)=\frac{x^2}{x^2+3}$ whose first and second derivatives are  $f'\left(x\right)=\frac{6x}{\left(x^2+3\right)^2}$ and  $f''\left(x\right)=\frac{18\left(1-x^2\right)}{\left(x^2+3\right)^3}$ respectively.

Use f'(x) to find any critical numbers. 

Consider the function $f\left(x\right)=\frac{x^2}{x^2+3}$ whose first and second derivatives are  $f'\left(x\right)=\frac{6x}{\left(x^2+3\right)^2}$ and  $f''\left(x\right)=\frac{18\left(1-x^2\right)}{\left(x^2+3\right)^3}$ respectively.

On what interval is f(x) increasing and concave down?

Given the graph of g, does the Mean Value Theorem apply over the interval [-7,-3]?

Given the graph of g, does the Mean Value Theorem apply on the interval $\left[-\pi,\pi\right]$  ?

Given the graph of f, does the Mean Value Theorem apply on the interval [3,6]?

The graph of f'(x) is shown. Which of the following statements is true about f(x)?

The graph of f'(x) is given.  On which of these intervals is f(x) decreasing?

Select all that apply:  If $f"\left(x\right)=x\left(x+1\right)\left(x-2\right)^2$  , then the graph of f has inflection points when x=

How many critical points does the function  $$f'\left(x\right)=\left(x+2\right)^5\left(x-3\right)^4$$  have?

If g is a differentiable function such that g(x)<0 for all real numbers x and if $f'\left(x\right)=\left(x^2-4\right)g\left(x\right)$  , which of the following is true?

For what value of k will $$f\left(x\right)=x+\frac{k}{x}$$  have a relative maximum at  $$x=-2$$  ?

Given the graph of f'(x), where is f(x) concave up?

Given the graph of f'(x), where does f(x) have an inflection point?  Select all that apply.

The function h is continuous and differentiable on the interval [-4,10]. The graph of h', the derivative of h, is shown in the graph.

On what interval(s) is h(x) decreasing?

The function h is continuous and differentiable on the interval [-4,10]. The graph of h', the derivative of h, is shown in the graph.

At which x-value(s) does h(x) has a relative maximum?

Let $$g'\left(x\right)=\left(x+1\right)^3\left(3-x\right)^2$$ .  At which x value(s) does g(x) have a relative minimum? 

Let $$g'\left(x\right)=\left(x+1\right)^3\left(3-x\right)^2$$ .  At which x value(s) does g(x) have a relative maximum? 

If y is a function of x such that y'>0 for all x and y"<0 for all x, which of the following could be part of the graph of y=f(x)?

The graph of the derivative of f is shown. Which of the following could be the graph of f?

Let f be a function that is continuous on the closed interval [-2,3] such that f'(0) does not exist, f'(2)=0, and f"(x)<0 for all x except x=0. Which of the following could be the graph of f?

The graph of a twice-differentiable function f is shown. Which of the following is true?

 If $$\left(j,k\right)$$ is a local minimum, then what will be true about $$f'(j)$$ ?

Determine the critical numbers of $$f\left(x\right)=3x^4-2x^3-9x^2$$

Let $$f\left(x\right)=\sqrt[]{x+9}$$ and let c be the number that satisfies the Mean Value Theorem on the interval [0,16]. What is c?

If the 1st derivative is positive on a certain interval, then that interval is decreasing.

If $$f'\left(2\right)=0$$ and $$f''\left(2\right)>0$$ , what can be said about f(x) at x=2?

If $$f'\left(2\right)=0$$ and $$f''\left(2\right)<0$$ , what can be said about f(x) at x=2?

If $$f'\left(2\right)=3$$ and $$f''\left(2\right)<0$$ , what can be said about f(x) at x=2?

Given a function g(x), if g''(x)=0 at a certain value of x, then g(x) must have _____________ at x.

The second derivative of the function f is given by $$f"\left(x\right)=x\left(x-a\right)\left(x-b\right)^2$$ .  The graph of f" is shown.  For what values of x does the graph of f have a point of inflection? 

Let f be a function with a second derivative given by $$f''\left(x\right)=x^2\left(x-3\right)\left(x-6\right)$$ .  What are the x-coordinates of the points of inflection of the graph of f? 

The function f has the property that $$f\left(x\right),\ f'\left(x\right),\ and\ f"\left(x\right)$$ are negative for all real values of x.  Which of the following could be the graph of f? 

Let f be the function given by $$f\left(x\right)=2xe^x$$ .  The graph is concave down when 

The function f is given by $$f\left(x\right)=x^3-2x^2$$ .  On what interval(s) if f(x) concave down? 

When calculating the absolute maximum of f on [a,b], which points need to be considered?

What is the first step to solve the problem below?

If $$f''\left(x\right)=x\left(x-1\right)\left(x-2\right)^2$$ , find the points of inflection for $$f\left(x\right)$$ .

How do you justify a point of inflection?

Which of these steps does not have to be done to find an absolute max of f(x)?

When is the graph of f(x) concave up?

Let f be the function defined by $$f\left(x\right)=3x^5-5x^3$$ .  How many relative extrema does f have? 

Let f be a function with first derivative by $$f'\left(x\right)=\frac{2x^3+x^2-4}{x^3+1}$$ for x>0.  It is known that f(0)=5 and f(3)=11.  What value of x in the open interval (0,3) satisfies the conclusion of the Mean Value Theorem for f on the closed interval [0,3]? 

Find the absolute max of $$y=\frac{2x^2}{3x+9}$$  over [-2,3]

Find the intervals of concavity for

f(x) = x² + 2x + 1.

How many inflection points does $$y=20x^5+\frac{10}{3}x^4$$  have?

If the function f has a critical number at $$x=3$$   and $$f''\left(3\right)=5$$ , which of the following would be true?

If the function f has a critical number at $$x=5$$   and $$f''\left(5\right)=-3$$ , which of the following would be true?

If $$f'\left(2\right)=4$$  and    $$f''\left(2\right)=-2$$ , which of the following would be true?

Given the table, for what values of x does the function f have a relative maximum?

Given the table, for what values of x does the function f have a relative minimum?

True or false: all critical points are extrema