Question

find the x - coordinates of all relative maxima of f(x).
f(x)=-\frac{3}{4}x^{4}-3x^{3}+105x^{2}+19

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime = nx^{n - 1}\), we have:
\(f^\prime(x)=-\frac{3}{4}\times4x^{3}-3\times3x^{2}+105\times2x+0=-3x^{3}-9x^{2}+210x=-3x(x^{2}+3x - 70)=-3x(x + 10)(x - 7)\)

Step2: Find the critical points

Set \(f^\prime(x) = 0\), then \(-3x(x + 10)(x - 7)=0\).
Solving \(x=0\), \(x=-10\), \(x = 7\)

Step3: Use the second - derivative test

First, find the second derivative. \(f^{\prime\prime}(x)=-9x^{2}-18x + 210\)

• For \(x=-10\):

\(f^{\prime\prime}(-10)=-9\times(-10)^{2}-18\times(-10)+210=-900 + 180+210=-510<0\)

• For \(x = 0\):

\(f^{\prime\prime}(0)=-9\times0^{2}-18\times0+210=210>0\)

• For \(x = 7\):

\(f^{\prime\prime}(7)=-9\times7^{2}-18\times7+210=-441-126 + 210=-357<0\)

Answer:

The \(x\) - coordinates of the relative maxima are \(x=-10\) and \(x = 7\)