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We can show that $f_r$ is injective without the derivative. Note that $$f_r(x)-f_r(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f_r(x)=f_r(y)$ if and only if $x=y$ because $$\underbrace{x^2+xy+y^2}_{\geq 0}+r>r\geq 0.$$

P.S. Injectivity is not influenced by the appearance of the inflection point! If $f'(x)\geq0$ for $x\in (a,b)$ and $f'(x)=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the MVT.

We can show that $f_r$ is injective without the derivative. Note that $$f_r(x)-f_r(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f_r(x)=f_r(y)$ if and only if $x=y$ because for $(x,y)\not=(0,0)$ $$\underbrace{x^2+xy+y^2}_{> 0}+r>r\geq 0.$$

P.S. Injectivity is not influenced by the appearance of an inflection point! If $f'(x)\geq0$ for $x\in (a,b)$ and $f'(x)=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the Mean Value Theorem.

We can show that $f_r$ is injective without the derivative. Note that $$f_r(x)-f_r(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f_r(x)=f_r(y)$ if and only if $x=y$ because $$\underbrace{x^2+xy+y^2}_{\geq 0}+r>r\geq 0.$$

P.S. Injectivity is not influenced by the appearance of the inflection point! If $f'(x)\geq0$ for $x\in (a,b)$ and $f'(x)=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the MVT.

We can show that $f_r$ is injective without the derivative. Note that $$f_r(x)-f_r(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f_r(x)=f_r(y)$ if and only if $x=y$ because $$\underbrace{x^2+xy+y^2}_{\geq 0}+r>r\geq 0.$$

P.S. Injectivity is not influenced by the appearance of the inflection point! If $f'(x)\geq0$ for $x\in (a,b)$ and $f'(x)=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the MVT.

We can show that $f$ is injective without the derivative. Note that $$f(x)-f(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f(x)=f(y)$ if and only f $x=y$ because $$\underbrace{x^2+xy+y^2}_{\geq 0}+r>r\geq 0.$$

We can show that $f_r$ is injective without the derivative. Note that $$f_r(x)-f_r(y)=(x-y)(x^2+xy+y^2+r).$$ Hence $f_r(x)=f_r(y)$ if and only if $x=y$ because $$\underbrace{x^2+xy+y^2}_{\geq 0}+r>r\geq 0.$$

P.S. Injectivity is not influenced by the appearance of the inflection point! If $f'(x)\geq0$ for $x\in (a,b)$ and $f'(x)=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the MVT.