Which of the following are correct?
(A) The function $f(x)=3x+12$ is increasing on R.
(B) The function $f(x) = e^{2x}$ is decreasing on R.
(C) The function $f(x) = x^2-x+1$ is neither increasing nor decreasing on (-1, 1).
(D) The function $f(x) = x^3-3x^2+4x$ is increasing on R.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
$\text{(A) }f(x)=3x+12\Rightarrow f'(x)=3,\ f'(x)>0\ \forall x\in\mathbb{R},\ \text{increasing on }\mathbb{R}\ (\text{Correct}).$
$\text{(B) }f(x)=e^{2x}\Rightarrow f'(x)=2e^{2x},\ f'(x)>0\ \forall x\in\mathbb{R},\ \text{increasing on }\mathbb{R}\ \text{(not decreasing)}\ (\text{Incorrect}).$
$\text{(C) }f(x)=x^{2}-x+1\Rightarrow f'(x)=2x-1.$
$\text{Critical point: }f'(x)=0\Rightarrow x=\frac{1}{2}.\ \text{On }(-1,\frac{1}{2}),\ f'(x)<0\ (\text{decreasing}),\ \text{on }(\frac{1}{2},1),\ f'(x)>0\ (\text{increasing}),\ \text{neither wholly increasing nor wholly decreasing on }(-1,1)\ (\text{Correct}).$
$\text{(D) }f(x)=x^{3}-3x^{2}+4x\Rightarrow f'(x)=3x^{2}-6x+4.$
$\text{Discriminant: }\Delta=(-6)^{2}-4\cdot3\cdot4=36-48=-12<0,\ \text{leading coefficient }3>0,\ f'(x)>0\ \forall x\in\mathbb{R},\ \text{increasing on }\mathbb{R}\ (\text{Correct}).$