For which of the following functions the second derivative test for finding extremum fails at x = 0 ?
For which of the following functions the second derivative test for finding extremum fails at x = 0 ?
The correct option is C f(x) = x3
If f’(c) = 0 , now according to the second derivative test, f”(c) < 0 => x = c is a maxima. If f”(c) > 0 , x = c is a minima. But if f”(c) is found to be zero we can’t say anything about it. The test fails.
We’ll apply second derivative test for each option.
a. f(x) = x2
f’(x) = 2x
f’(0) = 0
f”(x) = 2
Since f’(x) is positive we’ll have a minima at x = 0 for f(x) = x2
b. f(x) = x4
f'(x) = 4x3
f’(0) = 0
f”(x) = 12x2
f''(0) = 0
f”(x) is found to be zero, thus the test fails and further investigation is required.
c. f(x) = x3
f'(x) = 3x2
f’(0) = 0
f”(x) = 6x
f”(0) = 0
f”(x) is found to be zero, thus the test fails and further investigation is required.
f(x) = x3
If f’(c) = 0 , now according to the second derivative test, f”(c) < 0 => x = c is a maxima. If f”(c) > 0 , x = c is a minima. But if f”(c) is found to be zero we can’t say anything about it. The test fails.
We’ll apply second derivative test for each option.
a. f(x) = x2
f’(x) = 2x
f’(0) = 0
f”(x) = 2
Since f’(x) is positive we’ll have a minima at x = 0 for f(x) = x2
b. f(x) = x4
f'(x) = 4x3
f’(0) = 0
f”(x) = 12x2
f''(0) = 0
f”(x) is found to be zero, thus the test fails and further investigation is required.
c. f(x) = x3
f'(x) = 3x2
f’(0) = 0
f”(x) = 6x
f”(0) = 0
f”(x) is found to be zero, thus the test fails and further investigation is required.
