Find the Concavity f(x)=2x^3+3x^2-336x

Problem

ƒ(x)=2*x3+3*x2−336*x

Solution

1. Find the first derivative by applying the power rule to each term of the function.

d(ƒ(x))/d(x)=6*x2+6*x−336

2. Find the second derivative by differentiating the first derivative with respect to x

d2(ƒ(x))/(d(x)2)=12*x+6

3. Determine the inflection point by setting the second derivative equal to zero and solving for x

12*x+6=0

12*x=−6

x=−0.5

4. Test intervals to the left and right of x=−0.5 in the second derivative to determine the sign.

For *x<−0.5,ƒ″*(−1)=12*(−1)+6=−6

For *x>−0.5,ƒ(0)″=12*(0)+6=6

5. Identify concavity based on the sign of the second derivative: negative indicates concave down, and positive indicates concave up.

Concave down on *(−∞,−0.5)

Concave up on *(−0.5,∞)

Final Answer

Concave up on *(−0.5,∞), Concave down on *(−∞,−0.5)

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