6 short math problems

1.Let f(x)=1−4x1+4x.

Then f′(5) is _____

and f′′(5) is ______

2.Use the chain rule to find the derivative of

5(−9x^8+9x^6)^16

You do not need to expand out your answer.

3.Use the chain rule to find the derivative of

10sqrt(2x^3+6x^6)

Type your answer without fractional or negative exponents. Use sqrt(x) for x√.

4.Given the function g(x)=6x^3−9x^2−216x, find the first derivative, g′(x).

g′(x)= _______

Notice that g′(x)=0 when x=−3, that is, g′(−3)=0.

Now, we want to know whether there is a local minimum or local maximum at x=−3, so we will use the second derivative test.

Find the second derivative, g"(x).

g"(x)= _______

Evaluate g"(−3).

g"(−3)=______

Based on the sign of this number, does this mean the graph of g(x) is concave up or concave down at x=−3?

[Answer either up or down -- watch your spelling!!]

At x=−3 the graph of g(x) is concave _______

Based on the concavity of g(x) at x=−3, does this mean that there is a local minimum or local maximum at x=−3?

[Answer either minimum or maximum -- watch your spelling!!]

At x=−3 there is a local _________

5.The function f(x)=2x^3−39x^2+180x−10 has two critical numbers.

The smaller one is x =________

and the larger one is x = _________

6.The function f(x)=(5x+6)e^(−2x) has one critical number. Find it.

x = __________

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