ADDITIONAL EXERCISES FOR 5.1
Exercise 1. Recall that for any real number xthere is a unique integer nsuch
that n≤x < n + 1. The greatest integer function is given by [[x]] = n. Where are
the critical values of the greatest integer function? Which are local maxima and
which are local minima?
In exercises 2 through 5 find the local maxima and local minima of the given
function. For full credit find both the xand ycoordinates.
Exercise 2.
f(x) = ½x−1x < 2
x2x≥2
Exercise 3.
f(x) =
x−3x < 3
x33≤x≤5
1
xx > 5
Exercise 4.
f(x) = x2−98x+ 4
(Hint: Complete the square.)
Exercise 5.
f(x) = ½−2x= 0
1
x2x6= 0
Exercise 6. Explain why the function f(x) = 1
xhas no local maxima or minima.
All polynomials under consideration in the next three exercises have real variables
and real coefficients.
Exercise 7. How many critical points can a quadratic polynomial function have?
Exercise 8. Show that a cubic polynomial can have at most two critical points.
Give examples to show that a cubic polynomial may have zero, one, or two critical
points.
Exercise 9. We generalize the preceding two questions. Let nbe a p ositive integer
and let fbe a polynomial of degree n. How many critical points can fhave? (Hint:
use the fundamental theorem of algebra and Fermat’s theorem.)
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