Application of Derivatives · Mathematics · NDA

The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in the range $x>0$ is

What is $\varphi(a)$ equal to?

What is $\varphi'(a)$ equal to?

Which of the following is/are correct?

1. $f'(0) = 0$

2. $f''(0) < 0$

Select the correct answer using the code given below:

The function $y$ has a relative maxima at $x = 0$ for

Consider the following statements :

1. f(x) = In x is increasing in (0, ∞)

2. $g(x)=e^x+e^{\frac{1}{x}} $ is decreasing in (0, ∞)

Which of the statements given above is/are correct ?  

Consider the following statements in respect of the function f(x) = sin x:

1. f(x) increases in the interval (0, π).

2. f(x) decreases in the interval $\left(\dfrac{5\pi}{2},3\pi\right).$

Which of the above statements is/are correct?