Answer to Question #216461 in Calculus for ABUSALMA

Question

1)                 If f(x) = 3x2 – 2x + 5, find f[1, 2], f[2, 3] and f[1, 2, 3].

Answer 1

(1.) The function $f(x)=3x^2-2x+5$ is continuous for all $x\in \R$ as polynomial.

and the first derivative of this function f(x) is

$$f'(x)=6x-2$$ $$f'(x)=0\implies 6x-2=0\implies x=\dfrac{1}{3}$$

If $x>\dfrac{1}{3}, f'(x)>0, f(x)$ is strictly increasing.

$$f(1)=3(1)^2-2(1)+5=6$$ $$f(2)=3(2)^2-2(2)+5=13$$ $$f(3)=3(3)^2-2(3)+5=26$$ $$\implies f:[1,2]\to[6,13]$$ $$\implies f:[2,3]\to[13,26]$$ $$\implies f:\{1,2,3\}\to\{6, 13,26\}$$