What is the condition for local maxima using second derivative?
A
f''(x) = 0
B
f''(x) < 0
C
f''(x) > 0
D
f'(x) > 0
Analysis & Theory
If f'(x) = 0 and f''(x) < 0, then x is a point of local maximum.
What is the condition for local minima using second derivative?
A
f''(x) = 0
B
f''(x) < 0
C
f''(x) > 0
D
f'(x) > 0
Analysis & Theory
If f'(x) = 0 and f''(x) > 0, then x is a point of local minimum.
If f'(x) = 0 and f''(x) = 0, then:
A
Point of inflection
B
Local maximum
C
Local minimum
D
Test fails
Analysis & Theory
When f''(x) = 0, the second derivative test is inconclusive.
If f'(x₀) = 0 and f''(x₀) < 0, then:
A
x₀ is a point of local minimum
B
x₀ is a point of local maximum
C
x₀ is a saddle point
D
x₀ is an inflection point
Analysis & Theory
Negative second derivative indicates a concave-down curve, hence a local maximum.
If f'(x₀) = 0 and f''(x₀) > 0, then the function has:
A
No extremum
B
Point of local maximum
C
Point of local minimum
D
None of these
Analysis & Theory
Positive second derivative indicates concave-up, hence a local minimum.
If f(x) = x³, what can you say about x = 0 using second derivative test?
A
Local minimum
B
Local maximum
C
Test fails
D
Inflection point
Analysis & Theory
f'(0) = 0 and f''(0) = 0 → second derivative test fails at x = 0.
Which of the following is **not** a valid conclusion from second derivative test?
A
f''(x) > 0 ⇒ local minimum
B
f''(x) < 0 ⇒ local maximum
C
f''(x) = 0 ⇒ saddle point
D
f''(x) = 0 ⇒ test inconclusive
Analysis & Theory
f''(x) = 0 does not necessarily imply a saddle point; the test becomes inconclusive.
If f(x) = x², what is the nature of the critical point at x = 0?
A
Maximum
B
Minimum
C
Saddle point
D
None
Analysis & Theory
f'(0) = 0, f''(0) = 2 > 0 → x = 0 is a point of local minimum.
Which condition implies that a function is **concave upward** at a point?
A
f''(x) > 0
B
f''(x) < 0
C
f''(x) = 0
D
f'(x) = 0
Analysis & Theory
A positive second derivative implies the graph is concave upward.
If f(x) = -x², then what is the nature of x = 0?
A
Local minimum
B
Local maximum
C
Point of inflection
D
None
Analysis & Theory
f'(0) = 0 and f''(0) = -2 < 0 → x = 0 is a point of local maximum.