Given: A function-
[tex]f(x)=5-4x^2,\text{ }-5\leq x\leq1[/tex]
Required: To determine the absolute maxima and minima of the function.
Explanation: The given function is-
[tex]f(x)=5-4x^2[/tex]
Differentiating the function,
[tex]f^{\prime}(x)=-8x[/tex]
Setting f'(x)=0 gives-
[tex]\begin{gathered} -8x=0 \\ \Rightarrow x=0 \end{gathered}[/tex]
So we have to check the function at the boundary points of the interval [-5,1] and x=0 as follows-
Hence, the absolute maximum is 5 at x=o, and the minimum is -95 at x=-5.
Final Answer: The absolute maximum value is 5, and this occurs at x=0.
The absolute minimum value is -95, and this occurs at x=-5.
