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Problem 4.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.
Problem 4.1.67
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x²/₃(4 -x²) on -2,2
Problem 4.1.45
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³ - 3x² on [-1, 3]
Problem 4.1.43
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x² - 10 on [-2, 3]
Problem 4.1.23
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x² - 4x + 2
Problem 4.1.25
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ / 3 - 9x
Problem 4.1.27
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x³ + 3x² / 2 - 2x
Problem 4.1.29
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ -4a²x
Problem 4.1.31
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = t/ t² + 1
Problem 4.1.33
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = eˣ + e⁻ˣ
Problem 4.1.35
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 1 / x + ln x
Problem 4.1.37
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x² √(x + 5)
Problem 4.1.39
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x √(x-a)
Problem 4.1.41
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = 1/5 t⁵ - a⁴t
Problem 4.1.19
Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.
ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.
Problem 4.1.21
Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.
ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .
Problem 4.1.11
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
Problem 4.1.12
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
Problem 4.1.13
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
Problem 4.1.14
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
Problem 4.1.16
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.1.17
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.1.59
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = (4x³/3) + 5x² - 6x on [0,5]
Problem 4.1.61
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x/(x²+9)⁵ on [-2,2]
Problem 4.1.63
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = sec x on [-(π/4),π/4]
Problem 4.1.65
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³e⁻ˣ on [-1,5]
Problem 4.1.47
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]
Problem 4.1.49
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) cos² x on [0,π]
Problem 4.1.51
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = sin 3x on [-π/4,π/3]
Problem 4.1.71
Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?