For what values of x are the following functions differentia

For what values of x are the following functions differentiable? Explain your answer using definitions and theorems from this course. f(x) = x|x|, g(x) = |x -1 | + |x + 1| p(x) = x cos(1/x) q(x) = x^2 cos(1/x) Suppose f is differentiable at every x and |f(x) - f(y) |

Solution

7) Given that f(x) = x³+ 6x +8

To find the inverse, interchange x and y.

   x = y³+ 6y+8

   y³+ 6y = x -8

y( y²+6) = x -8

y = x -8 / (y²+6)  

Therefore,

   f^-1(x) = (x-8) / (y²+6)

Derivative of inverse function:

D[f-1()]= -8/ (y²+6)

6) Given function,

u_n = n³eⁿ²

   u_n+1 = (n+1)³ e⁽ⁿ⁺¹⁾² = (n+1)³ e^(n2+2n+1) [n2 = n²]

     un+1/un = (n+1)³ e^(n2+2n+1) / n³eⁿ²

  = (n+1/n)³ e^(n2+2n+1-n2)

= (1+1/n)³  e(2n+1)

Lt (un+1/un) =   Lt ( 1+1/n)³  e⁽²ⁿ⁺¹⁾

n--> n-->

= (1+1/)  e⁽²⁺¹⁾

   =

= diverges