Questions: Match the following descriptions to functions which meet that condition. A function, f(x) which is concave up for (a) f(x)=-x^2+3 all x values. A function, f(x) which is concave down (b) f(x)=cos(x) for all x values. A function, f(x), which is never concave (c) f(x)=e^x up or concave down (a function which has no concavity). A function, f(x), which has a point where (d) f(x)=2x+4 f'(x)=0 but no maximum or minimum values. A function, f(x), for which f'(x) will (e) f(x)=x^3 never equal 0. A function, f(x), with infinitely many (f) f(x)=3x maximums and minimums.

Solution Steps

To match the given descriptions to the functions, we need to analyze the properties of each function, such as concavity, points where the derivative is zero, and the behavior of the derivative.

1. For concavity, we look at the second derivative \( f''(x) \):

   • If \( f''(x) > 0 \), the function is concave up.
   • If \( f''(x) < 0 \), the function is concave down.
   • If \( f''(x) = 0 \), the function has no concavity.

2. For points where \( f'(x) = 0 \), we check if the function has any critical points and whether these points are maxima, minima, or neither.

3. For functions with infinitely many maxima and minima, we look for periodic functions.

1. Function \( f(x) = -x^2 + 3 \):

   • First derivative: \( f'(x) = -2x \)
   • Second derivative: \( f''(x) = -2 \)
   • Since \( f''(x) < 0 \) for all \( x \), the function is concave down for all \( x \).

2. Function \( f(x) = \cos(x) \):

   • First derivative: \( f'(x) = -\sin(x) \)
   • Second derivative: \( f''(x) = -\cos(x) \)
   • Since \( f''(x) \) can be positive or negative depending on \( x \), the function is concave up and concave down at different intervals.

3. Function \( f(x) = e^x \):

   • First derivative: \( f'(x) = e^x \)
   • Second derivative: \( f''(x) = e^x \)
   • Since \( f''(x) > 0 \) for all \( x \), the function is concave up for all \( x \).

4. Function \( f(x) = 2x + 4 \):

   • First derivative: \( f'(x) = 2 \)
   • Second derivative: \( f''(x) = 0 \)
   • Since \( f''(x) = 0 \), the function has no concavity (neither concave up nor concave down).

5. Function \( f(x) = x^3 \):

   • First derivative: \( f'(x) = 3x^2 \)
   • Second derivative: \( f''(x) = 6x \)
   • \( f'(x) = 0 \) at \( x = 0 \), but this is an inflection point, not a maximum or minimum.

6. Function \( f(x) = 3x \):

   • First derivative: \( f'(x) = 3 \)
   • Second derivative: \( f''(x) = 0 \)
   • Since \( f'(x) \) is a constant and never zero, the function has no maximum or minimum values.

1. A function, \( f(x) \), which is concave up for all \( x \) values:

   • \( f(x) = e^x \) (c)

2. A function, \( f(x) \), which is concave down for all \( x \) values:

   • \( f(x) = -x^2 + 3 \) (a)

3. A function, \( f(x) \), which is never concave up or concave down (a function which has no concavity):

   • \( f(x) = 2x + 4 \) (d)

Final Answer