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Question: Let f(x) = \(\frac{1}{3}\left\{ f(x + 1) + \frac{5}{f(x + 2)} \right\}\); then for all x, which one ...

Let f(x) = 13{f(x+1)+5f(x+2)}\frac{1}{3}\left\{ f(x + 1) + \frac{5}{f(x + 2)} \right\}; then for all x, which one is false?

A

f ′ is differentiable

B

f is differentiable

C

f ′ is continuous

D

f is continuous

Answer

f ′ is differentiable

Explanation

Solution

f(x) = {0;x<0x2;x≥0\left\{ \begin{array} { c l } 0 ; & x < 0 \\ x ^ { 2 } ; & x \geq 0 \end{array} \right.

f is continuous and differentiable

f ′(x) = {0;x<02x;x≥0\left\{ \begin{array} { c l } 0 ; & x < 0 \\ 2 \mathrm { x } ; & \mathrm { x } \geq 0 \end{array} \right.

f ′ continuous but not differentiable