Question

lf $x$ is real, then the maximum value of the expression $5+4x-4x^2$ is

• A. $6$
• B. $-6$
• C. $12$
• D. $-12$

Answer 1

The correct option is A $6$

$f\left(x\right)=-4x^2+4x+5$
Differentiating the above function with respect to x.
$f^{\prime }\left(x\right)=-8x+4$
$=0$
Or
$-8x+4=0$
$x=\frac{1}{2}$.
Now
$f^{\prime \prime }\left(x\right)=-8$
or
$f^{\prime \prime }(x{)}_{x=\frac{1}{2}}<0$
Hence the function attains a maximum value at x equal to $\frac{1}{2}$.
Therefore
$f\left(\frac{-1}{2}\right)=-4\left(\frac{1}{4}\right)+\frac{4}{2}+5$
$=-1+2+5$
$=6$.