# extreme values

## 2821 days ago by chrisphan

var('x') html("<h1>Example 1</h1>Below is a plot of $f(x) = x^3 - 3x^2 - 24x + 5$ (in red) and $f'(x) = 3x^2 - 6x - 24$ (in green).<br />The critical points are $x = -2$ and $x = 4$.<br />We see that the local extrema occur at critical points.") f = lambda x: x^3 - 3*x^2 - 24*x + 5 df = diff(f(x), x) show(plot(f, -4, 6, rgbcolor=(1, 0, 0)) + plot(df, -4, 6, rgbcolor=(0, 0.5, 0)) + point([[-2, 0], [4,0]], size=50))

# Example 1

Below is a plot of f(x) = x^3 - 3x^2 - 24x + 5 (in red) and f'(x) = 3x^2 - 6x - 24 (in green).
The critical points are x = -2 and x = 4.
We see that the local extrema occur at critical points.

# Example 1

Below is a plot of f(x) = x^3 - 3x^2 - 24x + 5 (in red) and f'(x) = 3x^2 - 6x - 24 (in green).
The critical points are x = -2 and x = 4.
We see that the local extrema occur at critical points.

var('x') html("<h1>Example 2</h1>Below is a plot of $g(x) = 3\sqrt[3]{x}(x^2 - 7)$ (in red) and $f'(x) = 7x^{-2/3}(x^2 - 1)$ (in green).<br />The critical points are $x = -1$, $x = 1$, and $x = 0$.<br />We see that the local extrema occur at critical points ($x = -1, 1$),<br /><b>but not every critical point yields a local extremum</b>:<br />there is no local extremum at $x = 0$.") g = lambda x: 3*sgn(x)*abs(x)^(1/3)*(x^2-7) dg = 7*(x^2 - 1)*abs(x)^(-2/3) show(plot(g, -1.5, 1.5, rgbcolor=(1, 0, 0)) + plot(dg, -1.5, 1.5, rgbcolor=(0, 0.5, 0)) + point([[-1, 0], [1,0]], size=50) , ymin=-20, ymax=20)

# Example 2

Below is a plot of g(x) = 3\sqrt[3]{x}(x^2 - 7) (in red) and f'(x) = 7x^{-2/3}(x^2 - 1) (in green).
The critical points are x = -1, x = 1, and x = 0.
We see that the local extrema occur at critical points (x = -1, 1),
but not every critical point yields a local extremum:
there is no local extremum at x = 0.

# Example 2

Below is a plot of g(x) = 3\sqrt[3]{x}(x^2 - 7) (in red) and f'(x) = 7x^{-2/3}(x^2 - 1) (in green).
The critical points are x = -1, x = 1, and x = 0.
We see that the local extrema occur at critical points (x = -1, 1),
but not every critical point yields a local extremum:
there is no local extremum at x = 0.

html("<h1>Example 3</h1>Below is a plot of $q(x) = |x|$ (in red) and $q'(x) = |x|/x$ (in green).<br />The only critical point is at $x = 0$.<br />We see that the local extrema occur at critical points.") q = lambda x: abs(x) dq = lambda x: abs(x)/x show(plot(q, -2, 2, rgbcolor=(1, 0, 0)) + plot(dq, -2, -0.01, rgbcolor=(0, 0.5, 0)) + plot(dq, 0.01, 2, rgbcolor=(0,0.5,0)))

# Example 3

Below is a plot of q(x) = |x| (in red) and q'(x) = |x|/x (in green).
The only critical point is at x = 0.
We see that the local extrema occur at critical points.

# Example 3

Below is a plot of q(x) = |x| (in red) and q'(x) = |x|/x (in green).
The only critical point is at x = 0.
We see that the local extrema occur at critical points.

html("<h1>Example 4</h1>Below is a plot of $r(x) = x^3 + 1$ (in red) and $r'(x) = 3x^2$ (in green).<br />The only critical point is at $x = 0$.<br />We see that there nevertheless is no local extrema at $x =0$.") q = lambda x: x^3 + 1 dq = lambda x: 3*x^2 show(plot(q, -1.75, 1.5, rgbcolor=(1, 0, 0)) + plot(dq, -1.75, 1.5, rgbcolor=(0, 0.5, 0)) + point([0, 0], size=50), ymin=-4, ymax = 4)

# Example 4

Below is a plot of r(x) = x^3 + 1 (in red) and r'(x) = 3x^2 (in green).
The only critical point is at x = 0.
We see that there nevertheless is no local extrema at x =0.

# Example 4

Below is a plot of r(x) = x^3 + 1 (in red) and r'(x) = 3x^2 (in green).
The only critical point is at x = 0.
We see that there nevertheless is no local extrema at x =0.

html("<h1>Example 5</h1>Below is a plot of $y = x - \cos x$ (in red) and $\\frac{dy}{dx} = 1+\sin x$ (in green) on $[0, 2\pi]$.<br />We see the minimum value is at $c = 0$ and the maximum value is at $c = 2\pi$.") y = lambda x: x- cos(x) dy = lambda x: 1 + sin(x) show(plot(y, 0, 2*pi, rgbcolor=(1, 0, 0)) + plot(dy, 0, 2*pi, rgbcolor=(0, 0.5, 0)) + point([[0, y(0)],[3*pi/2, y(3*pi/2)], [2*pi, y(2*pi)]], size=50))

# Example 5

Below is a plot of y = x - \cos x (in red) and \frac{dy}{dx} = 1+\sin x (in green) on [0, 2\pi].
We see the minimum value is at c = 0 and the maximum value is at c = 2\pi.

# Example 5

Below is a plot of y = x - \cos x (in red) and \frac{dy}{dx} = 1+\sin x (in green) on [0, 2\pi].
We see the minimum value is at c = 0 and the maximum value is at c = 2\pi.

html("<h1>Example 6</h1>Below is a plot of $y = x^3 -2x - 1/2$ (in red) and $\\frac{dy}{dx} = 3x^2 - 4$ (in green) on $[-1, 2]$.<br />We see the minimum value is at $c = \sqrt{2/3}$ and the maximum value is at $c = 2$.") y = lambda x: x^3 - 2*x - 1/2 dy = lambda x: 3*x^2 - 2 show(plot(y, -1, 2, rgbcolor=(1, 0, 0)) + plot(dy,-1, 2, rgbcolor=(0, 0.5, 0)) + point([[-1, y(-1)],[2, y(2)],[-sqrt(2/3), y(-sqrt(2/3))], [sqrt(2/3), y(sqrt(2/3))]], size=50), ymax=5, ymin =-2)

# Example 6

Below is a plot of y = x^3 -2x - 1/2 (in red) and \frac{dy}{dx} = 3x^2 - 4 (in green) on [-1, 2].
We see the minimum value is at c = \sqrt{2/3} and the maximum value is at c = 2.

# Example 6

Below is a plot of y = x^3 -2x - 1/2 (in red) and \frac{dy}{dx} = 3x^2 - 4 (in green) on [-1, 2].
We see the minimum value is at c = \sqrt{2/3} and the maximum value is at c = 2.