Consider the piecewise–defined function
f(x)={x2+2x+1,x2/3+3,−3≤x<0,0≤x≤4.
The domain for this problem is the closed interval [−3,4].
(a) Verify that f is continuous on [−3,4].
(b) Compute f′(x) on each subinterval where the derivative exists, and determine all interior points of [−3,4] at which f′(x)=0.
(c) Identify every interior point of [−3,4] at which f′(x) does not exist, and explain the geometric reason for the nondifferentiability at each such point.
(d) Create a complete list of x–values that must be tested to locate the absolute maximum and minimum values of f on [−3,4].
(e) Evaluate f at all points found in part (d) and determine the absolute maximum value and the absolute minimum value of f on [−3,4], stating where each occurs.