First Derivative Test for Local Minimum

Q. A commn tangent to 9x^2+16y^2=144 y^2=x-4 and x^2+y^2-12x+32=0 is

Q.

A differentiable function f(x) will have a local minimum at x = b if -

1. f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0

2. f’(b) = 0 , f’(b-h) > 0 & f’(b+h) < 0

3. f’(b) = 0

4. None of the above

Q. Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\prime \prime }\left(x\right)<0,\forall x\in (0,2).$ Then which of the following is (are) TRUE?

1. $g\left(x\right)$ is decreasing in $(0,\frac{4}{3})$
2. $g\left(x\right)$ is decreasing in $(\frac{4}{3},2)$
3. $g\left(\frac{1}{10}\right)<g\left(\frac{11}{10}\right)$
4. $g\left(x\right)$ has a local minimum at $x=\frac{4}{3}$