Quiz #4 Math 2210-12 James Wibowo¶

Given: \(\(f(x)=(x+3)(x^2-4)\)\)¶

1¶

Find the derivative \(f(x)=(x+3)(x^2-4)\) << FOIL \(f(x)=(x^3)+(-4x)+(3x^2)+(-12)\) \(f(x)=x^3+3x^2-4x-12\) << Find Derivative \(f'(x)=3x^2+6x^1-4+0\) \(\boxed{f'(x)=3x^2+6x-4}\)

2¶

Find the \(x\)-intercepts and the \(y\)-intercepts \(f(x)=x^3+3x^2-4x-12\) \(x\)-intercept \(0=x^3+3x^2-4x-12\) << Factor \(0=(x+3)(x+2)(x-2)\) \(\boxed{x=-3,-2,2}\) \(y\)-intercept \(f(x)=(x+3)(x^2-4)\) \(f(0)=(0+3)(0^2-4)\) \(f(0)=(3)(-4)\) \(f(0)=-12\) \(\boxed{y=-12}\)

3¶

Find the relative maximum and minimum \(f'(x)=3x^2+6x-4\) << Solve for \(x\)-intercept of derivative \(0=3x^2+6x-4\) \(x=0.527525316...,-2.52752523...\) Maximum \(y\) \(f(x)=(x+3)(x^2-4)\) \(f(x)=(-2.52752523+3)(-2.52752523^2-4)\) \(f(x)=1.12845108...\) \(\boxed{\textrm{Maximum: }(-2.528,1.128)}\) Minimum \(y\) \(f(x)=(x+3)(x^2-4)\) \(f(x)=(0.527525316+3)(0.527525316^2-4)\)

\(f(x)=-13.1284511...\)

\(\boxed{\textrm{Minimum: }(0.528,-13.128)}\)

4¶

Find the intervals, where the graph increases and decreases ie: down:\((-\infty,1)(3,\infty)\) \(\textrm{Up: }(-\infty,-2.528),(0.528,\infty)\) \(\textrm{Down: }(-2.528,0.528)\)

5¶

Find the intervals, where the graph is concave down and concave up \(f'(x)=3x^2+6x-4\) << Find derivative \(f''(x)=6x+6\) << Solve for \(x\)-intercept \(0=6x+6\) \(-6=6x\) \(x=-1\) \(\textrm{Concave down: }(-\infty,-1)\) \(\textrm{Concave up: }(-1,\infty)\)

6¶

Sketch the graph