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Visual Aids for Section 4.1, part 2The First Derivative Test
MTH 141
University of Rhode Island
MTH 141 (URI) Section 4.1 1 / 11
Our outcomes
Vocabulary: First Derivative Test for extrema.
Use the First Derivative Test to find and identify local maxima andminima of a given function.
Answer conceptual questions about the relationships among localextrema, critical points, and the signs of f ′(x) over intervals.
MTH 141 (URI) Section 4.1 2 / 11
How do you the local extrema of functions, given onlytheir formulas?
Our functions from the previous video, and a new one:
f (x) = x4 − 8x2 + 7
g(x) = arctan(x2)
h(x) = 4xe3x
j(x) = x4 − 6x2 − 8x
If f is defined on some interval (like a ≤ x ≤ b, for example), then anylocal maxima or local minima of f must occur at either critical pointsor at the interval endpoints.
Approach: We find local extrema in two steps:(1) Find the critical points by seeing where the first derivative is 0 or
undefined.(2) Decide whether each critical point is a local maximum, local
minimum, or neither. But how do we decide?
MTH 141 (URI) Section 4.1 3 / 11
How do you the local extrema of functions, given onlytheir formulas?
Our functions from the previous video, and a new one:
f (x) = x4 − 8x2 + 7
g(x) = arctan(x2)
h(x) = 4xe3x
j(x) = x4 − 6x2 − 8x
If f is defined on some interval (like a ≤ x ≤ b, for example), then anylocal maxima or local minima of f must occur at either critical pointsor at the interval endpoints.
Approach: We find local extrema in two steps:(1) Find the critical points by seeing where the first derivative is 0 or
undefined.(2) Decide whether each critical point is a local maximum, local
minimum, or neither. But how do we decide?
MTH 141 (URI) Section 4.1 3 / 11
How do you the local extrema of functions, given onlytheir formulas?
Our functions from the previous video, and a new one:
f (x) = x4 − 8x2 + 7
g(x) = arctan(x2)
h(x) = 4xe3x
j(x) = x4 − 6x2 − 8x
If f is defined on some interval (like a ≤ x ≤ b, for example), then anylocal maxima or local minima of f must occur at either critical pointsor at the interval endpoints.
Approach: We find local extrema in two steps:(1) Find the critical points by seeing where the first derivative is 0 or
undefined.(2) Decide whether each critical point is a local maximum, local
minimum, or neither. But how do we decide?
MTH 141 (URI) Section 4.1 3 / 11
How do you the local extrema of functions, given onlytheir formulas?
Our functions from the previous video, and a new one:
f (x) = x4 − 8x2 + 7
g(x) = arctan(x2)
h(x) = 4xe3x
j(x) = x4 − 6x2 − 8x
If f is defined on some interval (like a ≤ x ≤ b, for example), then anylocal maxima or local minima of f must occur at either critical pointsor at the interval endpoints.
Approach: We find local extrema in two steps:(1) Find the critical points by seeing where the first derivative is 0 or
undefined.(2) Decide whether each critical point is a local maximum, local
minimum, or neither.
But how do we decide?
MTH 141 (URI) Section 4.1 3 / 11
How do you the local extrema of functions, given onlytheir formulas?
Our functions from the previous video, and a new one:
f (x) = x4 − 8x2 + 7
g(x) = arctan(x2)
h(x) = 4xe3x
j(x) = x4 − 6x2 − 8x
If f is defined on some interval (like a ≤ x ≤ b, for example), then anylocal maxima or local minima of f must occur at either critical pointsor at the interval endpoints.
Approach: We find local extrema in two steps:(1) Find the critical points by seeing where the first derivative is 0 or
undefined.(2) Decide whether each critical point is a local maximum, local
minimum, or neither. But how do we decide?MTH 141 (URI) Section 4.1 3 / 11
One approach
The First-Derivative Test for Local Maxima and MinimaAfter finding the critical points of f , check the signs on f ′ around them. A
change from + to − means a local maximum, and a change from − to +
means a local minimum.
MTH 141 (URI) Section 4.1 4 / 11
One approach
The First-Derivative Test for Local Maxima and MinimaAfter finding the critical points of f , check the signs on f ′ around them. A
change from + to − means a local maximum, and a change from − to +
means a local minimum.
MTH 141 (URI) Section 4.1 4 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
f ′(−3) = −12(−1)(−5)
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
f ′(−1) = −4(1)(−3)
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
f ′(1) = 4(3)(−1)
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
f ′(1000000) = positive!
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
Find the local extrema of f (x) = x4 − 8x2 + 7.
From earlier: f ′(x) = 4x3 − 16x = 4x(x + 2)(x − 2)
Critical points at x = −2,0,2
f (x) has local minima at x = −2 and at x = 2, and a local maximum at x = 0.
MTH 141 (URI) Section 4.1 5 / 11
Example: First Derivative Test
f (x) = x4 − 8x2 + 7
MTH 141 (URI) Section 4.1 6 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
g′(−1) = −1 and g′(1) = 1
MTH 141 (URI) Section 4.1 7 / 11
Another example
Find the local extrema of g(x) = arctan(x2).
From earlier: g′(x) =2x
1 + x4
Critical point at x = 0
g(x) has local minimum at x = 0.
MTH 141 (URI) Section 4.1 7 / 11
Another example
g(x) = arctan(x2)
MTH 141 (URI) Section 4.1 8 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points?
x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
MTH 141 (URI) Section 4.1 9 / 11
A final example
Find the local extrema of j(x) = x4 − 6x2 − 8x .
Derivative: j ′(x) = 4x3 − 12x − 8 = 4(x − 2)(x + 1)2
Critical points? x = −1 and x = 2
j ′(−1000000) = negative j ′(0) = −8 j ′(1000000) = positive
j(x) has a local minimum at x = 2.
The point x = −1 is a critical point but not a local extremum.
MTH 141 (URI) Section 4.1 9 / 11
Another example
j(x) = x4 − 6x2 − 8x
MTH 141 (URI) Section 4.1 10 / 11
Our outcomes
Vocabulary: First Derivative Test for extrema.
Use the First Derivative Test to find and identify local maxima andminima of a given function.
Answer conceptual questions about the relationships among localextrema, critical points, and the signs of f ′(x) over intervals.
MTH 141 (URI) Section 4.1 11 / 11
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