Question

Find the concavity of the function given in the first picture, the second picture is the second derivative. Please use step by step solving method.

Answer 1

Step-by-step explanation:

The second derivative is already given, but here's how you can find it using product rule:

f(x) = x²e²ˣ

f'(x) = x² (2e²ˣ) + (2x) e²ˣ

f'(x) = 2e²ˣ (x² + x)

f"(x) = 2e²ˣ (2x + 1) + 4e²ˣ (x² + x)

f"(x) = 2e²ˣ (2x² + 4x + 1)

Now, we evaluate f"(x).  e²ˣ is always positive.  So let's check when the other factor is 0:

2x² + 4x + 1 = 0

x = [ -4 ± √(16 − 4(2)(1)) ] / 2(2)

x = [ -4 ± √(16 − 8) ] / 4

x = (-4 ± 2√2) / 4

x = (-2 ± √2) / 2

There are two places when f"(x) is 0.  So we can split the domain into 3 intervals:

x < (-2 − √2) / 2

(-2 − √2) / 2 < x < (-2 + √2) / 2

x > (-2 + √2) / 2

Evaluate the sign of f"(x) in each interval.

x < (-2 − √2) / 2 → f"(x) > 0

(-2 − √2) / 2 < x < (-2 + √2) / 2 → f"(x) < 0

x > (-2 + √2) / 2 → f"(x) > 0

So f(x) is concave down in the interval (-2 − √2) / 2 < x < (-2 + √2) / 2, and concave up everywhere else.