Section 4.1: Maximum and Minimum Values

1. 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, local minima at  2 and 4.     solution

2. Find the critical numbers of the function.

(a)   \(f(x)=x^3+x^2-x\)     solution

(b)   \(f(x)= x e^x\)     solution

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

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

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

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

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

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

(d)   \( f(x)=\dfrac{x^2-4}{x^2+4}, \quad [-4, 4]\)   solution

(e)   \( f(x) = x + \dfrac{1}{x}, \quad [0.5, 3] \)    solution