question\ngiven ( f(x)=x^{8}+4 x^{7}+10 ), find the ( x )-coordinates of all local minima using the second…

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (y = f(x)=x^{8}+4x^{7}+10), the first derivative (f^\prime(x)=8x^{7}+28x^{6}=4x^{6}(2x + 7))

Step2: Find the critical points

Set (f^\prime(x)=0). Then (4x^{6}(2x + 7)=0). (x^{6}=0) gives (x = 0), and (2x+7=0) gives (x=-\frac{7}{2})

Step3: Find the second derivative

Differentiate (f^\prime(x)=8x^{7}+28x^{6}) again. Using the power rule, (f^{\prime\prime}(x)=56x^{6}+168x^{5}=56x^{5}(x + 3))

Step4: Apply the second - derivative test

• For (x = 0): Substitute (x = 0) into (f^{\prime\prime}(x)), (f^{\prime\prime}(0)=56\times0^{5}(0 + 3)=0). The second - derivative test is inconclusive for (x = 0).
• For (x=-\frac{7}{2}): Substitute (x =-\frac{7}{2}) into (f^{\prime\prime}(x)). (f^{\prime\prime}\left(-\frac{7}{2}\right)=56\left(-\frac{7}{2}\right)^{5}\left(-\frac{7}{2}+3\right)) (=56\left(-\frac{16807}{32}\right)\left(-\frac{1}{2}\right)) (=56\times\frac{16807}{64}=\frac{56\times16807}{64}>0)

Answer:

(-\frac{7}{2})