Question

the function (f(x)=\frac{8}{sqrt{63}}x - arcsin(x)) has two critical values. determine these two critical values. enter the smaller critical value in the first blank and the larger critical value in the second blank. smaller critical value: larger critical value:

Explanation:

Step1: Find the derivative of the function

The function is \(y = \frac{8}{\sqrt{63}}x-\arcsin(x)\). The derivative \(y'=\frac{8}{\sqrt{63}}-\frac{1}{\sqrt{1 - x^{2}}}\) (using the derivative formulas \(\frac{d}{dx}(ax)=a\) and \(\frac{d}{dx}(\arcsin(x))=\frac{1}{\sqrt{1 - x^{2}}}\)).

Step2: Set the derivative equal to zero

Set \(\frac{8}{\sqrt{63}}-\frac{1}{\sqrt{1 - x^{2}}}=0\). Then \(\frac{1}{\sqrt{1 - x^{2}}}=\frac{8}{\sqrt{63}}\), cross - multiply to get \(\sqrt{63}=8\sqrt{1 - x^{2}}\). Square both sides: \(63 = 64(1 - x^{2})\). Expand: \(63=64 - 64x^{2}\). Rearrange to find \(64x^{2}=1\), so \(x^{2}=\frac{1}{64}\), and \(x=\pm\frac{1}{8}\).

Answer:

Smaller Critical Value: \(-\frac{1}{8}\)
Larger Critical Value: \(\frac{1}{8}\)