Let $f: R \rightarrow R$ given by $f(x)=\begin{cases} x^{5}+5x^{4}+10x^{3}+10x^{2}+3x+1, & x<0; \\ x^{2}-x+1, & 0 \leq x<1; \\ \frac{2}{3}x^{3}-4x^{2}+7x-\frac{8}{3}, & 1 \leq x<3; \\ (x-2)\log_{e}(x-2)-x+\frac{10}{3}, & x \geq 3 \end{cases}$ Then which of the following options is/are correct ?

Let f: R → R given by f(x)= x⁵+5x⁴+10x³+10x²+3x+1, x<0; x²-x+1, 0 ≤ x<1; 2/3x³-4x²+7x-8/3, 1 ≤ x<3; (x-2)logₑ(x-2)-x+10/3, x ≥ 3 Then which of the following options is/are correct ?

$f \text{ piecewise with breakpoints } x=0,\ x=1,\ x=3$

$\text{To find: which of } (1)\text{--}(4) \text{ are correct}$

$x<0:\ f(x)=x^5+5x^4+10x^3+10x^2+3x+1=(x+1)^5-2x$

$f(0^-)=1,\quad f(0^+)=1 \;\Rightarrow\; \text{continuous at } 0$

$f'(0^-)=3,\quad f'(0^+)=-1 \;\Rightarrow\; \text{not differentiable at } 0$

$f(1^-)=1,\quad f(1^+)=1 \;\Rightarrow\; \text{continuous at } 1$

$f'(1^-)=1,\quad f'(1^+)=1 \;\Rightarrow\; f'(1)=1$

$f(3^-)=f(3^+)=\dfrac{1}{3};\quad f\to-\infty \ (x\to-\infty),\ f\to+\infty \ (x\to+\infty)$

$\text{Plan: continuity/differentiability at } 0,1,3;\ \text{then the options}$

$f'(x)=5(x+1)^4-2;\quad f'(-1)=-2<0 \;\Rightarrow\; (2)\ \text{FALSE}$

$\text{Examine } f'' \text{ just left and right of } x=1$

$f''(1^-)=2,\quad f''(1^+)=-4 \;\Rightarrow\; f' \text{ has local max at } x=1 \;\Rightarrow\; (1)\ \text{TRUE}$

$f''(1^-)=2\neq-4=f''(1^+) \;\Rightarrow\; f' \text{ not differentiable at } 1 \;\Rightarrow\; (3)\ \text{TRUE}$

$f \text{ continuous},\ f(\mp\infty)=\mp\infty \;\Rightarrow\; \text{range}=\mathbb{R} \;\Rightarrow\; (4)\ \text{TRUE}$

$\text{Correct: } (1),\ (3),\ (4)$