Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: in particular, the last few exercises may be very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, or are independently marked.Derivatives and graphing1. X2 4 6y = f0(x)§ To the right, the graph of thederivative f0of a function f isshown. On what intervals is f in-creasing? Decreasing? Where is fconcave up? Concave down? Atwhat values does f have a localmaximum or minimum? Inflectionpoints?2. § Sketch the graph of a function f such that f0(1) = f0(−1) = 0, f0(x) < 0 if |x| < 1, f0(x) > 0if 1 < |x| < 2, f0(x) = −1 if |x| < 2, f00(x) < 0 if −2 < x < 0, and such that (0, 1) is aninflection point of y = f(x).3. § Suppose that f(x) = 2, f0(3) =12, and f0(x) > 0 and f00(x) < 0 for all x.(a) Sketch a possible graph of f.(b) How many solutions does the equation f(x) = 0 have?(c) Is it possible that f0(2) = 13?4. § For each of the following functions, find: intervals when f is increasing; intervals when f isdecreasing; intervals when f is concave up; intervals when f is concave down; local extremeof f; inflection points. Then sketch a graph of the function.(a) f(x) = 4x3+3x2−6x+1(b) f(x) = x2ln x(c) f(x) =√xe−x(d) f(x) = 200 + 8x3+ x4(c) f(x) = x1/3(x + 4)(d) f(x) = x + cos x5. Sketch a careful graph of y = (x2+ x + 1)ex. Label any interesting features (intercepts,asymptotes, extrema, points of inflection).6. (a) Let f(x) = (x − r)ex. Use calculus to sketch a graph of f (x), and label the zeros, localextrema, and inflection points. Also label the y-intercept and any horizontal asymptotes.(b) Let f(x) = (x2+ bx + c)ex. What is the behavior of f(x) as x → ±∞? Use the MeanValue Theorem to show that if f(x) has one or two zeros, then it must have two localextrema.1(c) More generally, let f(x) = p(x)ex, where p(x) is a polynomial of degree n. Show thatif f(x) has exactly n (real, distinct) zeros, then it also has exactly n local extrema andexactly n inflection points.(d) (Harder) Let’s return to the case when f (x) = (x2+ bx + c)ex. Prove that the zeros off correspond to the zeros of q0(x) = x2+ bx + c = f (x)/ex, and the number of these isclassified by the determinant b2− 4c.For n a non-negative integer, define qn(x) to be f(n)(x)/ex, the polynomial part of thenth derivative of ex. Prove that qn(x) is a quadratic for any n.How if the determinant of q1(x) related to the determinant of q0(x)? Is this the same asthe relationship between the determinants of qn(x) and qn+1(x) for arbitrary n? Whyor why not?Prove that for n large enough, qn(x), and hence f(n)(x), will have two roots.7. § Suppose the derivative of a function f is f0(x) = (x + 1)2(x −3)5(x −6)4. On what intervalsis f increasing? What are the local maxima of f?8. § Use calculus to sketch the family of curves y = x3− 3a2x + 2a3, where a is a positiveconstant.9. § Find the value of x such that f(x) =x + 1√x2+ 1increases most rapidly.10. § Find a cubic function f(x) = ax3+ bx2+ cx + d that has a local maximum value of 3 atx = −2 and a local minimum value of 0 at x = 1.11. § For what values of the numbers a and b does the function f(x) = axebx2have the maximumvalue f(2) = 1?12. § Show that the curve y =1 + x1 + x2has three points of inflection and that they all lie on onestraight line.13. § Show that the curves y = e−xand y = −e−xtouch the curve y = e−xsin x at its inflectionpoints.14. § Show that tan x > x for 0 < x < π/2. Hint: show that f(x) = tan x − x is increasing on(0, π/2).15. § Show that a cubic function always has precisely one point of inflection. Show that if thegraph has three x-intercepts x1, x2, and x3, then the x-coordinate of the inflection point is(x1+ x2+ x3)/2.What is the similar statement about local extrema of a quadratic function?16. (a) § Show that f(x) = x4is such that f00(0) = 0 but (0, 0) is not an inflection point of thegraph of f.(b) § Show that g(x) = x|x| has an inflection point at (0, 0) but g00(0) does not exist.(c) § Let f be any function. Use the First Derivative Test and Fermat’s Theorem on thefunction g = f0to show that if (c, f (c)) is an inflection point and f00exists in an openinterval that contains c, then f00(c) =