Section 4.1: Maximum and Minimum Values

Sketch an example of the graph as described below.

(a) Sketch the graph of a function \( f \) that is continuous on \([1,5]\) and has the absolute minimum at 2, absolute maximum at 5, local maximum at 3, and local minima at 2 and 4.  solution

(b) Sketch the graph of a function \( f \) that is continuous on \([-3,8]\), passes through the origin, has the absolute minimum at 7, absolute maximum at -3, local maximum at 3, and local minima at -1 and 7.  solution

(c) Sketch a function not continuous at 4 but with a local maximum at 4.  solution

(d) Sketch a function continuous on \([1,5]\) with absolute maximum at 3 but not differentiable at 3.  solution

Find the critical numbers of the function.

(a) \( f(x) = x^3 + x^2 - x \)  solution

(b) \( f(x) = x e^x \)  solution

(c) \( f(x) = x^3 e^{-5x} \)  solution

(d) \( f(x) = \frac{1}{3}x - x^{1/3} \)  solution

(e) \( f(t) = t^{3/4} - 6t^{1/4} \)  solution

Find the absolute maximum and minimum values of \( f \) on the interval.

(a) \( f(x) = 3x^2 - 12x + 5,\; [0,3] \)  solution

(b) \( f(x) = x^3 - 6x^2 + 9x + 2,\; [-2,2] \)  solution

(c) \( f(x) = x - \ln(5x),\; \left[\frac{1}{2}, 2\right] \)  solution

(d) \( f(x) = \frac{x}{x^2 + 1},\; [0,3] \)  solution

(e) \( f(x) = \frac{x^2 - 4}{x^2 + 4},\; [-4,4] \)  solution

(f) \( f(x) = x + \frac{1}{x},\; [0.5,3] \)  solution

(g) \( f(x) = x e^{2x},\; [-3,2] \)  solution