Second Derivative

Compute Second derivative

100

Find f′′(x) if

f(x)=x³

f′′(x) = 6x

200

Find f′′(x) if f(x) = sinx

f′′(x) = -sinx

300

Find f′′(x) if

f(x)=x⁴−2x²

f′′(x) = 12x² - 4 

400

Find f′′(x) if

f(x)=xe^x

f′(x)=e^x+xe^x=e^x(1+x) 

f′′(x)=e^x(1+x)+e^x=e^x(x+2)

500

f(x) = x^x

f(x)=e^xlnx 

f′(x)=x^x(ln⁡x+1)

f''(x)=x^x [(ln x+1)²+ 1/x]

Concavity and Inflection

100

If f′′(x)>0, is the function concave up or concave down?

If f′′(x)>0 → concave up

200

Is f(x)=x2f(x)=x^2f(x)=x2 concave up or concave down?

If f′′(x)>0 → concave up

300

Does f(x)=x³ have an inflection point at x=0?

Yes, inflection point at x=0

400

For f(x)=ln⁡x, determine concavity on x>0.

Concave down on x>0

500

Find the inflection points of

f(x)=x⁴−4x²

12x²-8 = 0

x² = 2/3

x = +/- sqrt(2/3)

Second Derivative Test

100

If f′(x₀)=0 and f''(x₀)>0, what type of critical point is x₀?

Local minimum

200

If f′(x₀)=0 and f''(x₀)<0, what type of point is it?

Local Max

300

If f′(x₀)=0 and f''(x₀)=0, what can we conclude?

→ Test is inconclusive

400

Classify the critical points of f(x)=x³−3x

f′′(1)=6>0⇒local minimum

f′′(−1)=−6<0⇒local maximum

 

500

Given f(x)=x⁴

Is x=0 a local minimum, maximum, or neither? Justify using the second derivative test.

Local minimum at x=0