Max & Min Problems (Leaving Cert Mathematics): Quizzes

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Max & Min Problems
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To find max or min of a function, what must you do with the derivative?

Set it equal to zero

For P(x)=−x2+10xP(x) = -x^2 + 10x, how many cakes give maximum profit?

5 cakes

What is the maximum profit when P(x)=−x2+10xP(x) = -x^2 + 10x and x=5x = 5?

25 euros

If perimeter 2l+2w=1002l + 2w = 100, express ww in terms of ll.

w=50−lw = 50 - l

For rectangle with w=50−lw = 50 - l, what is area in terms of ll?

50l−l250l - l^2

For Area=50l−l2Area = 50l - l^2, what value of ll maximises the area?

l=25l = 25

What are the optimal dimensions for the rectangular enclosure?

l=25l = 25, w=25w = 25

For cylinder with h=1000πr2h = \frac{1000}{\pi r^2}, what is C(r)C(r)?

2πr2+2000r2\pi r^2 + \frac{2000}{r}

What is C′(r)C'(r) for C(r)=2πr2+2000rC(r) = 2\pi r^2 + \frac{2000}{r}?

4πr−2000r24\pi r - \frac{2000}{r^2}

If C′′(r)>0C''(r) > 0 at a critical point, what does this indicate?

Local minimum

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The Basics

Curve Sketching

Implicit Differentiation

Rates of Change

Trig/Exponential Functions

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