Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x²
Point of local minimum: x = 0; Local minimum value = 0; No local maximum.
g(x) = x³ - 3x
Point of local maximum: x = -1, Local maximum value = 2; Point of local minimum: x = 1, Local minimum value = -2.
h(x) = sin x + cos x, 0 < x < π/2
Point of local maximum: x = π/4; Local maximum value = ; No local minimum.
f(x) = sin x - cos x, 0 < x < 2π
Point of local maximum: x = 3π/4, Local maximum value = ; Point of local minimum: x = 7π/4, Local minimum value = -.
f(x) = x³ - 6x² + 9x + 15
Point of local maximum: x = 1, Local maximum value = 19; Point of local minimum: x = 3, Local minimum value = 15.
g(x) = x/2 + 2/x, x > 0
Point of local minimum: x = 2; Local minimum value = 2; No local maximum.
g(x) = 1/(x² + 2)
Point of local maximum: x = 0; Local maximum value = 1/2; No local minimum.
f(x) = x, 0 < x < 1
Point of local maximum: x = 2/3; Local maximum value = 2/9; No local minimum.
Explanation
This question tests the application of first and second derivative tests to find local maxima and minima. For each function, we find critical points where f'(x) = 0, then use the second derivative test: if f''(c) > 0, it's a local minimum; if f''(c) < 0, it's a local maximum. The textbook context in Example 17 demonstrates this method for polynomial functions, and Definition 4 provides the theoretical foundation for identifying points of local maxima/minima.