Further Applications of Differentiation (AQA A-Level Mathematics): Quizzes

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Applications of the Second Derivative
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If f(x)>0f''(x) > 0 for all xx in an interval, what does this indicate about f(x)f(x)?

f(x)f(x) is concave up on that interval

When f(x)<0f''(x) < 0 on an interval, what happens to the first derivative?

The first derivative is decreasing

How do you find potential inflection points?

Solve f(x)=0f''(x) = 0 or where f(x)f''(x) is undefined

For a local minimum at x=cx = c, what conditions must be satisfied?

f(c)=0f'(c) = 0 and f(c)>0f''(c) > 0

For a local maximum at x=cx = c, what must f(c)f''(c) satisfy?

f(c)<0f''(c) < 0

Given f(x)=x33x2+2f(x) = x^3 - 3x^2 + 2 and f(x)=6x6f''(x) = 6x - 6, what is f(2)f''(2)?

66

When is the second derivative test inconclusive?

When f(x)=0f'(x) = 0 and f(x)=0f''(x) = 0

For maximum profit, what must the profit function P(x)P(x) satisfy?

P(x)<0P''(x) < 0 (concave down)

What does the second derivative of position s(t)s(t) represent in physics?

Acceleration a(t)a(t)

A function is convex if which condition holds for all xx in its domain?

f(x)0f''(x) \geq 0

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