f:[0, 3]→b, f(x)=2x^3-15x^2+36x+7 is an onto function g:[0, ∞]→d, g(x)=x2025/x2025+1 is also an onto function.

Question

\(f:[0,3] \rightarrow b, f(x)=2 x^3-15 x^2+36 x+7\) is an onto function \(g:[0, \infty) \rightarrow d, g(x)=\frac{x^{2025}}{x^{2025}+1}\) is also an onto function. Find the number of elements in the set \(S=\{x: x \in Z, x \in b or \ x \in d\}\)

Answer 1

Answer is "30"  

\(f(x)=2 x^3-15 x^2+36 x+7\)

\(f^{\prime}(x)=6 x^2-30 x+36=0 \)

\(\Rightarrow \quad x^2-5 x+6=0 \)

\(\therefore x=1,5 \)

\(f(0)=7, f(2)=35, f(3)=34 \)

\(\therefore b=[7,35] \)

\(g(x)=\frac{x^{2025}}{1+x^{2025}} \)

d = [0, 1)

\(\therefore S=[0,7,8,9, \ldots, 35]\)

Number of elements = 30