Let f(x) = x^4(x - 1)^3. (a) Find the critical numbers of the function f (Enter your answers from smallest to largest: ) smallest value X1 X2 largest value X3 (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? At X1 the second derivative test indicates local minimum At X2 the second derivative test indicates neither minimum nor maximum At X3 the second derivative test indicates local maximum (C) What does the First Derivative Test tell you? Note what the First Derivative Test tells you that Second Derivative Test does not: At X1 the first derivative test indicates local minimum At X2 the first derivative test indicates neither minimum nor maximum At X3 the first derivative test indicates local maximum
Added by Morgan S.
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Step 1: Find the critical numbers of the function f by setting the first derivative equal to 0: \(f'(x) = 7x^6 - 36x^5 + 60x^4 - 32x^3\) Setting \(f'(x) = 0\), we get: \(7x^6 - 36x^5 + 60x^4 - 32x^3 = 0\) This equation gives us critical numbers at \(x = 0\), \(x Show more…
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Recall that the Second Derivative Test has the following basic steps (see text and slides for more details) 1. Find all critical numbers for the function using the FIRST derivative. 2. Evaluate each critical number in the SECOND derivative. 3. Use the sign of what you get in Step 2 to determine the concavity of the function at a critical number. 4. Use the concavity to determine if you have a local max, local min, or an inconclusive situation. (Recall the happy and sad faces) Now... Consider the function f(x) = x3 – 8x² + 19x – 12 . Use the SECOND DERIVATIVE TEST TO DO THE FOLLOWING. A) Find the two critical numbers for this function. Round each one to nearest tenth (one decimal place). Enter them separated by commas: Ans = B) Start with the SMALLER critical number (which may be negative). What is the value of the second derivative at that number? Be sure to use the rounded value you entered above to do you computation. If you need the quadratic formula, see this [LINK]. Ans = C) Based on the result in Part B, we can conclude there is a Local maximum at the smaller critical value. We know this because the second derivative is negative at that point and that tells us the graph is concave down at that point. D) Now start with the LARGER critical number. What is the value of the second derivative at that number? Be sure to use the rounded value you entered above to do you computation. Ans = E) Based on the result in Part D, we can conclude there is a Local minimum at the smaller critical value. We know this because the second derivative is postive at that point and that tells us the graph is concave up at that point.
Kathleen C.
Perform a first derivative test on the function f(x) = 2x^3 + 3x^2 - 36x + 9; [ - 3,6]. a. Locate the critical points of the given function. b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). a. Locate the critical points of the given function. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The critical point(s) is/are at x = (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no critical points. b. Locate the local maximum value. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. There is a local maximum at x = (Type an integer or a simplified fraction.) B. There is no local maximum. Locate the local minimum value. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. There is a local minimum at x = (Type an integer or a simplified fraction.) B. There is no local minimum. c. Identify the absolute maximum value of the function. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The absolute maximum value is (Type an integer or a simplified fraction.) B. There is no absolute maximum.
Adi S.
Let f(x) = x^4(x - 1)^3. (a) Find the critical numbers of the function f. (Enter your answers from smallest to largest.) smallest value x1 = x2 = largest value x3 = (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? At x1 the second derivative test ---Select--- At x2 the second derivative test ---Select--- At x3 the second derivative test ---Select--- (c) What does the First Derivative Test tell you? Note what the First Derivative Test tells you that Second Derivative Test does not. At x1 the first derivative test ---Select--- At x2 the first derivative test ---Select--- At x3 the first derivative test ---Select---
Sri K.
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