Local maxima and local minima

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

• There is a local max at x=3
• There is a local min at x = 3
• There is an inflection point at x = 3
• There is an x-intercept at x = 3

If

 $f\left(x\right)=\frac{1}{4x^2+2x+1},$  then its maximum value is _____

• 3/4

• 4/3

• 1

• none of these

The function $f\left(x\right)=\sin x+\cos x,\ 0<x<\frac{\pi}{2}$ is  

• local maximum at  $x=\frac{\pi}{4}$  

• local minimum at $x=\frac{\pi}{4}$  

• none of these

The function $f(x)=x^3\ –\ 6x^2+9x+15$ is local maxima at 

• x = 1

• x = 3

• no point of maxima and minima

• none of these

The function $f\left(x\right)=\frac{x}{2}+\frac{2}{x},\ \ x>0$ is 

• local maximum at x = 2

• local minimum at x = 2

• none of these

The function $f(x)=x^3\ –\ 3x+3$ is 

• local maximum at x = - 1

• local minimum at x = - 1

• local maximum at x = 1

• none of these

For minimum point (i)  $\frac{\text{d}y}{\text{d}x}=0$  
(ii)  $\frac{\text{d}y}{\text{d}x}$  changes sign from -ve to +ve 

• False

• True