4.19 Exercises, p. 191

Problem 1

f[x_] := (x^2) - 3 x + 2;
f'[x]

f''[x]

Plot[f[x], {x, 0, 3}]

Problem 2

f[x_] := (x^3) - 4 x;
f'[x]

f''[x]

Plot[f[x], {x, -5, 5}]

Problem 3

\(f[x_] := \((x - 1)\)\^2\ \((x + 2)\); 
\(f'\)[x]\)Factor[f'[x]]

f''[x] // Simplify

Plot[f[x], {x, -3, 3}]

Problem 4

f[x_] := (x^3) - 6 (x^2) + 9 x + 5;
f'[x]f'[x] // Factor

f''[x]

Plot[f[x], {x, 0, 4}]

Problem 5

\(f[x_] := 2 + \((x - 1)\)\^4; 
\(f'\)[x]\)

f''[x]

Plot[f[x], {x, -1, 3}, PlotRange -> {0, 10}]

Problem 6

f[x_] := (1/(x^2));
f'[x]

f''[x]

Plot[f[x], {x, -2, 2}, PlotRange -> {0, 10}]

Problem 7

f[x_] := x + (1/(x^2));
f'[x]

f''[x]

Plot[f[x], {x, -2, 2}, PlotRange -> {-5, 10}]

Problem 8

f[x_] := (1/((x - 1) (x - 3)));
f'[x]//Simplify

f''[x]//Simplify

Plot[f[x], {x, -1, 5}, PlotRange -> {-5, 5}]

Problem 9

f[x_] := (x/(1 + (x^2)));
f'[x]//Simplify

f''[x]//Simplify

Plot[f[x], {x, -4, 4}]

Problem 10

f[x_] := (((x^2) - 4)/((x^2) - 9));
f'[x]//Simplify

f''[x]//Simplify

Plot[f[x], {x, -4, 4}, PlotRange -> {-5, 5}]

Problem 11

f[x_] := Sin[x]^2;

f[x] == 0 iff x == n Pi, for some integer n.

f'[x]

f'[x] == 0 iff x == n Pi/2, for some integer n.

f''[x]

f''[x] == 0 iff Tan[x] == +-1 iff x == n Pi/4, for some odd integer n.

Plot[f[x], {x, -2 π, 2 π}]

Problem 12

f[x_] := x - Sin[x];
f'[x]

f'[x]==0 iff x == 2n Pi for some integer n.

f''[x]

f''[x]==0 iff x == n Pi for some integer n.

Plot[f[x], {x, -3 π, 3 π}]

Problem 13

f[x_] := x + Cos[x];
f'[x]

f'[x]==0 iff x == Pi/2 + 2n Pi for some integer n.

f''[x]

f''[x]==0 iff x == n Pi/2 for some odd integer n.

Plot[f[x], {x, -3 π, 3 π}]

Problem 14

f[x_] := ((x^2)/6) + (1/12) Cos[2 x];
f'[x]

Plot[{(x/3), (1/6) Sin[2 x]}, {x, -π, π}]

f'[x]==0 iff x==0

f''[x]

f''[x]==0 iff x = n Pi for some integer n.
Note: these points are NOT inflection  points because the sign of f''[x] does not change (f''[x]>=0).

Plot[f[x], {x, -3 π, 3 π}]