Amit M Agarwal Solutions for Chapter: Maxima and Minima, Exercise 3: Target Exercises

If $h\left(x\right)=f\left(x\right)+f\left(-x\right)$, then $h\left(x\right)$ has got an extreme value at a point, where $f^{\text{'}}\left(x\right)$ is

The least value of $a$ for which the equation, $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has at least one solution on the interval $\left(0,\frac{\pi }{2}\right)$, is

The minimum value of  $2^{{\left(x^2-3\right)}^3+27}$ is

If a differentiable function $f\left(x\right)$ has a relative minimum at $x=0,$ then the function $y=f\left(x\right)+ax+b$ has a relative minimum at $x=0$ for

$f\left(x\right)={\int }_{\cos x}^{\sin x}\left(1-t+2t^3\right)dt$ has in $\left[0,2\pi \right]$,

The function $f\left(x\right)=\frac{x}{1+x\tan x}$ has

The denominator of a fraction is greater than $16$ of the square of the numerator, then the least value of the fraction is

Let $f\left(x\right)=a{x}^3+5x^2+cx+1$ be a polynomial function. If $f\left(x\right)$ has extreme at $x=\alpha$ and $\beta$ such that $\alpha \beta <0$ and $f\left(\alpha \right)f\left(\beta \right)<0$, then the equation $f\left(x\right)=0$ has