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AI HL · AHL 2

Further Derivatives

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AHL 2.7

The second derivative and concavity

AHL 2.7

The second derivative and concavity

Target: I can find the second derivative of a polynomial, an exponential or a ln function and evaluate it at a point. (+4 more)

By the end of this section

Key points

The second derivative

  • Differentiate f′(x) to get the second derivative f″(x). It is the rate of change of the gradient.
  • In the other notation, \(\frac{dy}{dx}\) is the derivative and \(\frac{d^2y}{dx^2}\) is the second derivative.
  • Differentiate twice, then substitute. Substituting after the first derivative gives f′, not f″.
  • The derivative of \(e^{kx}\) is \(ke^{kx}\), so the second derivative is \(k^2e^{kx}\). The derivative of \(\ln x\) is \(\frac1x\), so its second derivative is \(-\frac{1}{x^2}\).

Concave-up and concave-down

  • Where f″(x) > 0 the gradient is increasing and the curve is concave-up (it bends like a cup).
  • Where f″(x) < 0 the gradient is decreasing and the curve is concave-down (it bends like a cap).
  • Increasing or decreasing comes from f′. Concave-up or concave-down comes from f″. A section can be any one of the four pairs, such as decreasing and concave-up.
  • To find where a curve is concave-down, solve f″(x) < 0. Concave-up: solve f″(x) > 0.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Second derivative of a cubic

If \(f(x)=2x^3-5x^2+4x\), find \(f''(3)\).

Working
  1. First derivative

    \(f'(x)=6 x^{2} - 10 x + 4\)

    Differentiate each term: the power comes down and drops by one.

  2. Second derivative

    \(f''(x)=12 x - 10\)

    Differentiate f′ in the same way. The constant term goes.

  3. Substitute

    \(f''(3)=12\times 3-10=26\)

    Only now put in x = 3.

Video walkthrough coming
Second derivative of an exponential

If \(y=2e^{-3x}\), find \(y''(1)\) to 3 significant figures.

Working
  1. First derivative

    \(y'=- 6 e^{- 3 x}\)

    The derivative of \(e^{kx}\) is \(ke^{kx}\): multiply by −3.

  2. Second derivative

    \(y''=18 e^{- 3 x}\)

    Multiply by −3 again. Two negatives give a positive.

  3. Substitute

    \(y''(1)=18e^{-3}=0.896\)

    Use the calculator for the value, then give 3 significant figures.

Video walkthrough coming
Second derivative of a ln function

If \(y=5\ln x+2\), find \(y''(3)\) to 3 significant figures.

Working
  1. First derivative

    \(y'=\frac{5}{x}=5x^{-1}\)

    The derivative of ln x is 1/x. The constant 2 goes. Write 5/x as a power to differentiate again.

  2. Second derivative

    \(y''=-5x^{-2}=-\frac{5}{x^2}\)

    Bring down −1, and the power becomes −2.

  3. Substitute

    \(y''(3)=-\frac{5}{9}=-0.556\)

    A ln curve is concave-down for every x > 0: y″ is negative.

Video walkthrough coming
Where is a curve concave-down?

Find the values of x for which \(f(x)=2x^3+9x^2-5x+3\) is concave-down.

Working
  1. Second derivative

    \(f'(x)=6x^2+18x-5,\quad f''(x)=12x+18\)

    Concave-down means f″(x) < 0.

  2. Inequality

    \(12x+18<0\)

    Solve like an equation. Dividing by a positive number keeps the sign.

  3. Answer

    \(x<-1.5\)

    Left of −1.5 the gradient is decreasing: the curve is concave-down. Right of −1.5 it is concave-up.

Video walkthrough coming
Describing sections of a graph

The graph shows \(f(x)=x^3-9x^2+24x\). Use the words increasing, decreasing, concave-up and concave-down to describe (a) A to B, (b) C to E.

1234565101520xyABCDE
Working
  1. Gradient

    A to B: the curve rises. C to D: it falls. D to E: it rises.

    Read increasing or decreasing from the direction of the curve, left to right.

  2. Bend

    \(f''(x)=6x-18\)

    f″ is zero at x = 3, which is point C. Left of C it is negative (concave-down). Right of C it is positive (concave-up).

  3. (a) A to B

    Increasing, concave-down.

    The gradient is positive and getting smaller.

  4. (b) C to E

    C to D: decreasing, concave-up; D to E: increasing, concave-up.

    The curve is concave-up all the way, but it turns from falling to rising at D.

Video walkthrough coming