Question

If the derivative of a continuous function does not change its sign on its entire domain, then the function will be one-one.

• A. one-one-ento
• B. f(x)=1+sqrt(x), f:[0,inf) -> [0,inf)
• C. f(x)=2x^3-3x^2+4x-1, f:R -> R
• D. f(x)=x^3-7x^2+2x+2025, f:R -> R
• E. f(x)=(3x-1)/(x+2); f:R-{2}->R
• F. f(x)=x^3-x^2-x+4+(sinx), f:R->R

Answer 1

The statement is correct. A continuous function whose derivative does not change sign on its domain is strictly monotonic, and therefore one-to-one. For the given examples:

1. $f(x)=1+\sqrt{x}$ on $[0,\infty)$ has $f'(x) = \frac{1}{2\sqrt{x}} > 0$ for $x>0$. It is one-to-one but not onto as its range is $[1,\infty)$ while codomain is $[0,\infty)$.
2. $f(x)=2x^3-3x^2+4x-1$ on $R$ has $f'(x) = 6x^2-6x+4$. The discriminant is $(-6)^2 - 4(6)(4) = 36-96 = -60 < 0$. Since the leading coefficient is positive, $f'(x) > 0$ for all $x \in R$. It is strictly increasing and thus one-to-one. As a cubic polynomial, its range is $R$, so it is also onto.
3. $f(x)=x^3-7x^2+2x+2025$ on $R$ has $f'(x) = 3x^2-14x+2$. This quadratic has real roots, so $f'(x)$ changes sign. Thus, $f(x)$ is not one-to-one.
4. $f(x)=\frac{3x-1}{x+2}$ on $R-\{2\}$ has $f'(x) = \frac{7}{(x+2)^2} > 0$ for $x \neq -2$. It is one-to-one.
5. $f(x)=x^3-x^2-x+4+(\sin x)$ on $R$ has $f'(x) = 3x^2-2x-1+\cos x$. $f'(0)=0$ and $f''(0)=-2$, indicating $f'(x)$ changes sign. Thus, $f(x)$ is not one-to-one.

Answer 2

A function is one-to-one if its derivative does not change sign over its domain.

1. $f(x)=1+\sqrt{x}$: $f'(x) = \frac{1}{2\sqrt{x}} > 0$ for $x>0$. One-to-one.
2. $f(x)=2x^3-3x^2+4x-1$: $f'(x) = 6x^2-6x+4$. Discriminant $\Delta < 0$ and leading coefficient $>0$, so $f'(x) > 0$. One-to-one.
3. $f(x)=x^3-7x^2+2x+2025$: $f'(x) = 3x^2-14x+2$. Roots exist, derivative changes sign. Not one-to-one.
4. $f(x)=\frac{3x-1}{x+2}$: $f'(x) = \frac{7}{(x+2)^2} > 0$ for $x \neq -2$. One-to-one.
5. $f(x)=x^3-x^2-x+4+(\sin x)$: $f'(x) = 3x^2-2x-1+\cos x$. $f'(0)=0$ and $f''(0)=-2$, so derivative changes sign. Not one-to-one.

The functions that are one-to-one are ①, Ⓐ, and Ⓑ.