this is section 3.3 problem 8: the function y = h(x) is represented by the graph in the textbook. use the…

this is section 3.3 problem 8: the function y = h(x) is represented by the graph in the textbook. use the graph to find intervals and x - values that satisfy the following conditions. the answers should be in ascending order and either be an integer or an integer and a half such as \1.5\. use \dne\ if such interval does not exist. (a) the intervals on which the derivative is positive are ( ), ( ) and ( ). (b) the intervals on which the derivative is negative are ( ), ( ) and ( ). (c) the x - values at which the derivative equals zero are x = , and . (d) the x - values at which the derivative does not exist are x = and . hint: refer to theorem 1.

this is section 3.3 problem 8: the function y = h(x) is represented by the graph in the textbook. use the graph to find intervals and x - values that satisfy the following conditions. the answers should be in ascending order and either be an integer or an integer and a half such as \1.5\. use \dne\ if such interval does not exist. (a) the intervals on which the derivative is positive are ( ), ( ) and ( ). (b) the intervals on which the derivative is negative are ( ), ( ) and ( ). (c) the x - values at which the derivative equals zero are x = , and . (d) the x - values at which the derivative does not exist are x = and . hint: refer to theorem 1.

Answer

Explanation:

Step1: Identify positive - derivative intervals

The derivative of a function is positive when the function is increasing. Looking at the graph of (y = h(x)), the function is increasing on the intervals where the slope of the tangent line is positive. We can see that the function is increasing on the intervals ((-3,-2)) and ((1,3)).

Step2: Identify negative - derivative intervals

The derivative of a function is negative when the function is decreasing. From the graph, the function is decreasing on the intervals ((-2,1)) and ((3,4)).

Step3: Identify zero - derivative values

The derivative of a function is zero at the local maxima and minima (where the tangent line is horizontal). From the graph, the derivative is zero at (x=-2) and (x = 1).

Step4: Identify non - existent derivative values

The derivative does not exist at points of sharp corners or vertical tangents. From the graph, the derivative does not exist at (x=-3) and (x = 4) (assuming the graph has sharp endpoints).

Answer:

(a) ((-3,-2)), ((1,3)) (b) ((-2,1)), ((3,4)) (c) (-2), (1) (d) (-3), (4)