Lesson 21 (4.1)

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Lesson 21 (4.1)

Today: Maxima and Minima

Office Hours: MWF: 130pm - 230pm, MATH 842.

Announcements: * Exam 2 on Monday (10/19)

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Warmup Example:

What are the maximum & minimum values of the functions graphed below

Two Cartesian coordinate graphs shown side-by-side. The left graph is labeled f(x), defined on the interval [-1, 4]. It starts at an endpoint (-1, 1), rises to a smooth local peak at (0, 3), falls through the x-axis to a local minimum at (2, -1), and then increases sharply to a closed endpoint at (4, 5). Dashed lines mark the coordinates for (-1, 1), (2, -1), and (4, 5). The right graph is labeled g(x), defined on the interval [-4, 7]. It starts at an endpoint near x = -4, rises smoothly to a global peak/maximum at (-2, 6), descends across the y-axis down to a sharp V-shaped cusp minimum at (1, -3), and then rises with a slight inflection to end at a closed endpoint at (7, 5). Dashed lines indicate the coordinates for (-2, 6), (1, -3), and (7, 5).
Visual Description: Two Cartesian coordinate graphs shown side-by-side. The left graph is labeled f(x), defined on the interval [-1, 4]. It starts at an endpoint (-1, 1), rises to a smooth local peak at (0, 3), falls through the x-axis to a local minimum at (2, -1), and then increases sharply to a closed endpoint at (4, 5). Dashed lines mark the coordinates for (-1, 1), (2, -1), and (4, 5). The right graph is labeled g(x), defined on the interval [-4, 7]. It starts at an endpoint near x = -4, rises smoothly to a global peak/maximum at (-2, 6), descends across the y-axis down to a sharp V-shaped cusp minimum at (1, -3), and then rises with a slight inflection to end at a closed endpoint at (7, 5). Dashed lines indicate the coordinates for (-2, 6), (1, -3), and (7, 5).

\(\max = 5\) at \(x = 4\)
\(\min = -1\) at \(x = 2\)

\(\max = 6\) at \(x = -2\)
\(\min = -3\) at \(x = 1\)

occur at end points or at places where \(\text{Derivative} = 0\) or Does not exist.

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Critical points:

\( f(x) \) is a continuous function

\( x = c \) is called a critical point if

\[ f'(c) = 0 \quad \text{or} \quad f'(c) \text{ DNE} \]

Extreme Value Theorem:

\( f(x) \) is continuous on a closed interval \( [a, b] \)

then \( f(x) \) has an absolute max & absolute min and they occur either at end points \( x = a \), \( x = b \) or at critical points

Evaluate function at \( x = a \), \( x = b \) & critical points, and compare

the Largest \( = \) Max, smallest \( = \) min.

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e.g.:

\[ f(x) = x^3 - 3x^2 + 1 \quad \text{over } [-1, 4] \]

End points: \( x = -1, \quad x = 4 \)

Critical points: \( f'(x) = 3x^2 - 6x = 0 \implies x = 0 \text{ and } x = 2 \)

No points where derivative does not exist.

So \( x = 0 \) & \( x = 2 \) are the only critical points.

\( f(-1) = -1 - 3 + 1 = -3 \)
\( f(0) = 0 - 0 + 1 = 1 \)
\( f(2) = 8 - 12 + 1 = -3 \)
\( f(4) = 64 - 48 + 1 = 17 \)

Largest \( = 17 \)
\( \Rightarrow \text{Max} = 17 \) at \( x = 4 \)

Smallest \( = -3 \)
\( \Rightarrow \text{min} = -3 \) at \( x = -1 \text{ & } x = 2 \)

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Hand-drawn graph of a continuous cubic curve on a grid over the domain [-1, 4]. The horizontal x-axis is marked with ticks at x = -1, 2, and 4. The vertical y-axis is labeled at y = 17, 1, and -3. The curve starts at the point (-1, -3), rises to a local maximum on the y-axis at (0, 1), turns downward crossing the x-axis to reach a local minimum at (2, -3), and then rises steeply crossing the x-axis again to reach an endpoint at (4, 17). A dashed horizontal guideline at y = -3 connects the points (-1, -3), (0, -3), and (2, -3). Another dashed horizontal guideline at y = 17 connects the y-axis at 17 to the point (4, 17).
Visual Description: Hand-drawn graph of a continuous cubic curve on a grid over the domain [-1, 4]. The horizontal x-axis is marked with ticks at x = -1, 2, and 4. The vertical y-axis is labeled at y = 17, 1, and -3. The curve starts at the point (-1, -3), rises to a local maximum on the y-axis at (0, 1), turns downward crossing the x-axis to reach a local minimum at (2, -3), and then rises steeply crossing the x-axis again to reach an endpoint at (4, 17). A dashed horizontal guideline at y = -3 connects the points (-1, -3), (0, -3), and (2, -3). Another dashed horizontal guideline at y = 17 connects the y-axis at 17 to the point (4, 17).

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Example: Find Absolute Max. & Min.

e.g.:

\[ f(x) = 3x^4 - 16x^3 + 18x^2 \quad \text{over } [-1, 4] \]

End points:

\( x = -1, \quad x = 4 \)

Critical:

\[ f'(x) = 12x^3 - 48x^2 + 36x = 0 \] \[ 12x(x^2 - 4x + 3) = 0 \] \[ 12x(x - 1)(x - 3) = 0 \]

3 critical points \( x = 0, \, x = 1, \, x = 3 \)

Compare:

\[ f(-1) = 3 + 16 + 18 = 37 \implies \text{Max} \] \[ f(0) = 0 \] \[ f(1) = 3 - 16 + 18 = 5 \] \[ f(3) = 3(3)^4 - 16(3^3) + 18(3^2) = 3^2 [3^3 - 16 \times 3 + 18] = 3^2 [27 - 48 + 18] = -27 \implies \text{Min.} \] \[ f(4) = 3(4)^4 - 16(4)^3 + 18(4)^2 = 16 [48 - 64 + 18] = 32 \]

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Hand-drawn Cartesian coordinate plane on grid paper showing a continuous blue polynomial curve. The horizontal x-axis is labeled with tick marks at -1, 0, 1, 3, and 4. The vertical y-axis is labeled at -27, 0, 5, 32, and 37. Key features of the curve include: a left endpoint at (-1, 37) with a horizontal dashed line extending to y = 37; the curve decreases to touch the origin (0, 0); it rises to a local maximum at (1, 5) with dashed lines indicating x = 1 and y = 5; it decreases, crossing the positive x-axis, to reach a minimum at (3, -27) with a dashed line to y = -27; it turns and rises steeply, crossing the x-axis again between 3 and 4, ending at a point at (4, 32) with a dashed horizontal line connecting across to y = 32.
Visual Description: Hand-drawn Cartesian coordinate plane on grid paper showing a continuous blue polynomial curve. The horizontal x-axis is labeled with tick marks at -1, 0, 1, 3, and 4. The vertical y-axis is labeled at -27, 0, 5, 32, and 37. Key features of the curve include: a left endpoint at (-1, 37) with a horizontal dashed line extending to y = 37; the curve decreases to touch the origin (0, 0); it rises to a local maximum at (1, 5) with dashed lines indicating x = 1 and y = 5; it decreases, crossing the positive x-axis, to reach a minimum at (3, -27) with a dashed line to y = -27; it turns and rises steeply, crossing the x-axis again between 3 and 4, ending at a point at (4, 32) with a dashed horizontal line connecting across to y = 32.

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Example

\[ f(x) = \sqrt{2}\cos x + \sin^2 x \quad \text{over } [0, \pi/2] \]

End points: \( x = 0, \quad x = \pi/2 \)

Critical:

\[ f'(x) = -\sqrt{2} \sin x + 2 \sin x \cos x = 0 \]

\[ \sin x [-\sqrt{2} + 2\cos x] = 0 \]

\[ \sin x = 0 \quad \text{OR} \quad \cos x = \frac{\sqrt{2}}{2} \]

  • For \( \sin x = 0 \):
    \( x = 0, \pi, -\pi, 2\pi, \dots \)
    Only \( x = 0 \) is in the required interval
  • For \( \cos x = \sqrt{2}/2 \):
    \( x = \pi/4, 7\pi/4, -\pi/4, \dots \)
    \( x = \pi/4 \) is the only critical point

Compare:

\[ f(0) = \sqrt{2} \approx 1.414 \]

\[ f(\pi/4) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + \left(\frac{\sqrt{2}}{2}\right)^2 = 1 + 1/2 = 1.5 \quad \text{max} \]

\[ f(\pi/2) = 1 \quad \text{min} \]

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\( f(x) = \sqrt{2}\cos x + \sin^2 x \quad \text{on} \quad [-\pi/2, \pi/2] \)

end points: \( x = -\pi/2 \), \( x = \pi/2 \)

critical points:

\[ f'(x) = -\sqrt{2}\sin x + 2\sin x \cos x = 0 \]

\[ \sin x = 0 \quad \text{or} \quad \cos x = \frac{\sqrt{2}}{2} \]

For \( \sin x = 0 \):
\( x = 0 \) is the only point in \( [-\pi/2, \pi/2] \)

For \( \cos x = \frac{\sqrt{2}}{2} \):
\( x = \pi/4 \) & \( x = -\pi/4 \) are two points in \( [-\pi/2, \pi/2] \)

Evaluating at the critical points and endpoints:

\[ f(-\pi/2) = 1 \] \[ f(-\pi/4) = \sqrt{2} * \frac{\sqrt{2}}{2} + \left(\frac{\sqrt{2}}{2}\right)^2 = 1.5 \] \[ f(0) = \sqrt{2} = 1.414 \] \[ f(\pi/4) = 1.5 \] \[ f(\pi/2) = 1 \]

Conclusion:

Max = \( 1.5 \) at \( x = \pi/4 \) & \( -\pi/4 \)

Min = \( 1 \) at \( -\pi/2 \) & \( \pi/2 \)

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e.g: Find absolute max. & min. for \( f(x) = 2\sin^{-1}x - 2x^2 \) over \( [-1, 1] \)

end points: \( x = -1, \quad x = 1 \)

critical:

\[ f'(x) = \frac{2}{\sqrt{1-x^2}} - 4x = 0 \implies \frac{2}{\sqrt{1-x^2}} = 4x \] \[ \frac{1}{1-x^2} = 4x^2 \] \[ 1 = 4x^2(1-x^2) \] \[ = 4x^2 - 4x^4 \] \[ 4x^4 - 4x^2 + 1 = 0 \] \[ (2x^2)^2 - 2(2x^2)(1) + 1^2 = 0 \] \[ (2x^2 - 1)^2 = 0 \]

\[ x = \frac{1}{\sqrt{2}} \quad \text{or} \quad x = -\frac{1}{\sqrt{2}} \]

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\( f(x) = 2 \sin^{-1}x - 2x^2 \)

\[ f(-1) = 2 \sin^{-1}(-1) - 2 = 2(-\pi/2) - 2 = -\pi - 2 \approx -5.14 \]

\[ f(-1/\sqrt{2}) = 2 \cdot \sin^{-1}(-1/\sqrt{2}) - 2(1/2) = 2(-\pi/4) - 1 = -\pi/2 - 1 \approx -2.57 \]

\[ f(1/\sqrt{2}) = 2 \sin^{-1}(1/\sqrt{2}) - 2(1/2) = 2(\pi/4) - 1 = \pi/2 - 1 \approx 0.57 \]

\[ f(1) = 2 \sin^{-1}(1) - 2 = 2(\pi/2) - 2 = \pi - 2 \approx 1.14 \]

Hand-drawn Cartesian graph showing sine/arcsine coordinate mapping on grid paper. The horizontal axis includes markings at -\pi/2, -\pi/4, \pi/4, and \pi/2. The vertical axis includes markings at 1, 1/\sqrt{2}, -1/\sqrt{2}, and -1. Dashed projection lines connect the points (\pi/4, 1/\sqrt{2}), (\pi/2, 1), (-\pi/4, -1/\sqrt{2}), and (-\pi/2, -1) along a sketched curve.
Visual Description: Hand-drawn Cartesian graph showing sine/arcsine coordinate mapping on grid paper. The horizontal axis includes markings at -\pi/2, -\pi/4, \pi/4, and \pi/2. The vertical axis includes markings at 1, 1/\sqrt{2}, -1/\sqrt{2}, and -1. Dashed projection lines connect the points (\pi/4, 1/\sqrt{2}), (\pi/2, 1), (-\pi/4, -1/\sqrt{2}), and (-\pi/2, -1) along a sketched curve.

\( \text{max} = -5.14 \text{ at } x = -1 \)

\( \text{min} = 1.14 \text{ at } x = 1. \)

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e.g. Find absolute max. & min. for \( f(x) = x + \frac{1}{x} \) over \( \left[\frac{1}{2}, 2\right] \)

end points: \( x = \frac{1}{2} \), \( x = 2 \)

Critical:

\[ f'(x) = 1 - \frac{1}{x^2} = 0 \implies x^2 = 1 \implies x = 1, \; x = -1 \] \[ f'(x) = 1 - \frac{1}{x^2} \quad \text{DNE at} \quad x = 0 \]

\( x = 1 \) is the only critical in \( \left[\frac{1}{2}, 2\right] \)

\[ f\left(\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{1/2} = 2.5 \qquad \max = 2.5 \text{ at } x = \frac{1}{2} \ \& \ x = 2 \] \[ f(1) = 1 + 1 = 2 \qquad \min = 2 \text{ at } x = 1 \] \[ f(2) = 2 + \frac{1}{2} = 2.5 \]