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

  3. (a) Sketch the graph of a function f that is not continuous at 4 but has a local maximum at 4.   solution
    (b) Sketch the graph of a function f that is continuous on [1, 5] and has the absolute maximum at 3 but f is not differentiable at 3.   solution

  4. Find the critical numbers of the function.

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

    (b)   \(f(x)= x e^x\)     solution

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

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

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

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

    6. (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)   \( f(x)=\dfrac{x}{x^2+1}, \quad [0, 3]\)  solution

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

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