Question

example3
given the graph of a function ( f ) below, approximate the maximum and minimum values of ( f ) and its location on the given interval.
\ta. (1,2) \t\t\t\t\tb. (-1,2)
\tc. (-1,0) \t\t\t\t\td. (-2,1)

Explanation:

To solve for the maximum and minimum values of \( f \) on each interval, we analyze the graph:

Part a: Interval \([1, 2]\)

• Step 1: Analyze the graph on \([1, 2]\)

The function is decreasing on \([1, 2]\) (from \( x = 1 \) to \( x = 2 \)).

• Step 2: Find maximum and minimum
• Maximum: At \( x = 1 \), the value is \( 2 \) (from the peak near \( x = 1 \)).
• Minimum: At \( x = 2 \), the value is \( 0 \) (crosses the x - axis at \( x = 2 \)).

Part b: Interval \([-1, 2]\)

• Step 1: Identify critical points

The graph has a minimum at \( x=-1 \) (value \( - 2 \)) and a peak at \( x = 1 \) (value \( 2 \)), and ends at \( x = 2 \) (value \( 0 \)).

• Step 2: Compare values
• Maximum: At \( x = 1 \), \( f(1)=2 \) (highest point in \([-1, 2]\)).
• Minimum: At \( x=-1 \), \( f(-1)=-2 \) (lowest point in \([-1, 2]\)).

Part c: Interval \([-1, 0]\)

• Step 1: Analyze the graph on \([-1, 0]\)

The function is increasing on \([-1, 0]\) (from \( x=-1 \) to \( x = 0 \)).

• Step 2: Find maximum and minimum
• Maximum: At \( x = 0 \), the value is \( 1 \) (y - intercept).
• Minimum: At \( x=-1 \), the value is \( - 2 \) (lowest point in \([-1, 0]\)).

Part d: Interval \([-2, 1]\)

• Step 1: Identify critical points

The graph has a minimum at \( x=-2 \) (value \( - 2 \)) and a peak at \( x = 1 \) (value \( 2 \)).

• Step 2: Compare values
• Maximum: At \( x = 1 \), \( f(1)=2 \) (highest point in \([-2, 1]\)).
• Minimum: At \( x=-2 \), \( f(-2)=-2 \) (lowest point in \([-2, 1]\)).

Final Answers:

• a. \([1, 2]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( 0 \) at \( x = 2 \).

• b. \([-1, 2]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( - 2 \) at \( x=-1 \).

• c. \([-1, 0]\):

Maximum: \( 1 \) at \( x = 0 \); Minimum: \( - 2 \) at \( x=-1 \).

• d. \([-2, 1]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( - 2 \) at \( x=-2 \).

Answer:

To solve for the maximum and minimum values of \( f \) on each interval, we analyze the graph:

Part a: Interval \([1, 2]\)

• Step 1: Analyze the graph on \([1, 2]\)

The function is decreasing on \([1, 2]\) (from \( x = 1 \) to \( x = 2 \)).

• Step 2: Find maximum and minimum
• Maximum: At \( x = 1 \), the value is \( 2 \) (from the peak near \( x = 1 \)).
• Minimum: At \( x = 2 \), the value is \( 0 \) (crosses the x - axis at \( x = 2 \)).

Part b: Interval \([-1, 2]\)

• Step 1: Identify critical points

The graph has a minimum at \( x=-1 \) (value \( - 2 \)) and a peak at \( x = 1 \) (value \( 2 \)), and ends at \( x = 2 \) (value \( 0 \)).

• Step 2: Compare values
• Maximum: At \( x = 1 \), \( f(1)=2 \) (highest point in \([-1, 2]\)).
• Minimum: At \( x=-1 \), \( f(-1)=-2 \) (lowest point in \([-1, 2]\)).

Part c: Interval \([-1, 0]\)

• Step 1: Analyze the graph on \([-1, 0]\)

The function is increasing on \([-1, 0]\) (from \( x=-1 \) to \( x = 0 \)).

• Step 2: Find maximum and minimum
• Maximum: At \( x = 0 \), the value is \( 1 \) (y - intercept).
• Minimum: At \( x=-1 \), the value is \( - 2 \) (lowest point in \([-1, 0]\)).

Part d: Interval \([-2, 1]\)

• Step 1: Identify critical points

The graph has a minimum at \( x=-2 \) (value \( - 2 \)) and a peak at \( x = 1 \) (value \( 2 \)).

• Step 2: Compare values
• Maximum: At \( x = 1 \), \( f(1)=2 \) (highest point in \([-2, 1]\)).
• Minimum: At \( x=-2 \), \( f(-2)=-2 \) (lowest point in \([-2, 1]\)).

Final Answers:

• a. \([1, 2]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( 0 \) at \( x = 2 \).

• b. \([-1, 2]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( - 2 \) at \( x=-1 \).

• c. \([-1, 0]\):

Maximum: \( 1 \) at \( x = 0 \); Minimum: \( - 2 \) at \( x=-1 \).

• d. \([-2, 1]\):

Maximum: \( 2 \) at \( x = 1 \); Minimum: \( - 2 \) at \( x=-2 \).