Let a be a real number such that the function f ( x ) = a x 2 + 6 x - 15 , x ∈ R is increasing in -…

We have,

$f\left(x\right)=a{x}^2+6x-15$

Graph of $f\left(x\right)$ is shown below as it is increasing in $\left(-\infty ,\frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4},\infty \right)$.

Then, abscissa of vertex is $\frac{3}{4}$.

$\frac{-B}{2A}=\frac{3}{4}$

$\Rightarrow \frac{-\left(6\right)}{2a}=\frac{3}{4}$

$\Rightarrow a=-4$

$\therefore g\left(x\right)=-4x^2-6x+15$

Local maxima exists at the vertex of $g\left(x\right)$, whose abscissa is

$x=\frac{-B}{2A}$

$\Rightarrow x=-\left(\frac{\left(-6\right)}{-8}\right)=-\frac{3}{4}$