Compilation of Solved Problems in Differential Calculus
Maxima and Minima
First Derivative Test
1) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 4 is the only critical number of f.
Testing:
x < 4 | f'(0) = -8 | f is decreasing | x > 4 | f'(5) = 2 | f is increasing |
By the First Derivative Test, x = 4 is a local minimum.
2) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -3 and 6 are the only critical numbers of f.
Testing:
x < -3 | f'(-10) = 672 | f is increasing | -3 < x < 6 | f(0) = -108 | f is decreasing | x >6 | f(10) = 312 | f is increasing |
By the First Derivative Test, x = -3 is a local minimum and x = 6 is a local maximum.
3) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 0 and 0.75 are the only critical numbers of f.
Testing:
x < 0 | f'(-1) = -7 | f is decreasing | 0 < x < 0.75 | f(0.5) = -0.25 | f is decreasing | x >0.75 | f(1) = 1 | f is increasing |
By the First Derivative Test, x = 1 is a local minimum.
4) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -2 and 0 are the only critical numbers of f.
Testing:
x < -2 | f'(-10) = 80 e- 10 | f is increasing | -2 < x < 0 | f(-1) = - e- 1 | f is decreasing | x > 0 | f(1) = 3 e | f is increasing |
By the First Derivative Test, x = - 2 is a local minimum and x = 0 is a local maximum.
5)We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 0 is the only critical number of
First Derivative Test
1) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 4 is the only critical number of f.
Testing:
x < 4 | f'(0) = -8 | f is decreasing | x > 4 | f'(5) = 2 | f is increasing |
By the First Derivative Test, x = 4 is a local minimum.
2) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -3 and 6 are the only critical numbers of f.
Testing:
x < -3 | f'(-10) = 672 | f is increasing | -3 < x < 6 | f(0) = -108 | f is decreasing | x >6 | f(10) = 312 | f is increasing |
By the First Derivative Test, x = -3 is a local minimum and x = 6 is a local maximum.
3) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 0 and 0.75 are the only critical numbers of f.
Testing:
x < 0 | f'(-1) = -7 | f is decreasing | 0 < x < 0.75 | f(0.5) = -0.25 | f is decreasing | x >0.75 | f(1) = 1 | f is increasing |
By the First Derivative Test, x = 1 is a local minimum.
4) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -2 and 0 are the only critical numbers of f.
Testing:
x < -2 | f'(-10) = 80 e- 10 | f is increasing | -2 < x < 0 | f(-1) = - e- 1 | f is decreasing | x > 0 | f(1) = 3 e | f is increasing |
By the First Derivative Test, x = - 2 is a local minimum and x = 0 is a local maximum.
5)We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 0 is the only critical number of