Let $f(x)=|x^2-3x-4|$ defined for $-1\le x\le 4$. Which of the following is correct?

Factorize the expression inside the modulus: $g(x)=x^2-3x-4=(x-4)(x+1)$.For $-1\le x\le 4$, the product $(x-4)(x+1)$ is $\le 0$.Thus, $f(x)=|x^2-3x-4|=-(x^2-3x-4)=-x^2+3x+4$.To find the maximum value, find critical points:$f^{\prime }(x)=-2x+3=0⟹x=1.5$.Since $f^{\prime \prime }(x)=-2<0$, $x=1.5$ is a point of local maxima.The maximum value is $f(1.5)$:
$$f(1.5)=-(1.5^2-3(1.5)-4)=-(2.25-4.5-4)=-(-6.25)=6.25$$
Checking boundaries: $f(-1)=0$ and $f(4)=0$.So the maximum value is $6.25$.