derivatives

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows: $$ f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right) $$ Which of the following statements is/are true?

Let $$f(x) = {x^3} + 15{x^2} - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?

Consider the functions I. $${e^{ - x}}$$ II. $${x^2} - \sin x$$ III. $$\sqrt {{x^3} + 1} $$ Which of the above functions is/are increasing everywhere in [0,1]?

Let $$f(x)$$ be a polynomial and $$g\left( x \right) = f'\left( x \right)$$ be its derivative. If the degree of $$\left( {f\left( x \right) + f\left( { - x} \right)} \right)$$ is $...

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve $$3{x^4} - 16{x^3} + 24{x^2} + 37$$ is

Find the points of local maxima and minima if any of the following function defined in $$0 \le x \le 6,$$ $$\,\,\,\,f\left( x \right) = {x^3} - 6{x^2} + 9x + 15.$$

What is the maximum value of the function $$f\left( x \right) = 2{x^2} - 2x + 6$$ in the interval $$\left[ {0,2} \right]$$?