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![(i) Verify the Mean Value Theorem for
y=x^(3)-3x+1 on the interval
[-2,2] as follows.
(a) Verify that the hypotheses of the Mean value theorem holds.
(b) Verify that the conclusion of the Mean value theorem holds.
(ii) Illustrate (i-(b)) by a graph involving the extreme chord.](https://blblobstorage.blob.core.windows.net/solver-b2c/1ade063a-002b-11ef-bc53-f2bdc25d22fe.jpg)
(i) Verify the Mean Value Theorem for on the interval as follows.(a) Verify that the hypotheses of the Mean value theorem holds.(b) Verify that the conclusion of the Mean value theorem holds.(ii) Illustrate (i-(b)) by a graph involving the extreme chord.
Full solution
Q. (i) Verify the Mean Value Theorem for on the interval as follows.(a) Verify that the hypotheses of the Mean value theorem holds.(b) Verify that the conclusion of the Mean value theorem holds.(ii) Illustrate (i-(b)) by a graph involving the extreme chord.
- Check Continuity: Check if the function is continuous on the closed interval . is a polynomial, which is continuous everywhere.
- Check Differentiability: Check if the function is differentiable on the open interval . is a polynomial, which is differentiable everywhere.
- Calculate Average Rate: Calculate the average rate of change of the function on the interval .Average rate of change =
- Find Derivative: Find the derivative of the function.
- Set Derivative Equal: Set the derivative equal to the average rate of change to find the value in .Since we are looking for a value in the interval , we take .
- Verify Value of c: Verify that this value of is within the interval . is approximately , which is between and .
- Draw Graph: Draw the graph to illustrate the conclusion of the Mean Value Theorem. The graph should show the function , the secant line connecting the points and , and the tangent line at that is parallel to the secant line.
