# F. Pleae help with a step by step guide. Thank you. Let f(x)=x^3 −12x + 5(a)  What is the y-intercept?(b)  Find all critical points for the first derivative.(c)  Use the second derivative test to determine whether these critical points are minima, maxima, or points of inflection.(d)  Use the second derivative to find any points of inflection.(e)  Sketch the graph, showing and labeling all extrema, points of inflection, and the y-intercept. You do not need to determine the x-intercepts.

Question

F. Pleae help with a step by step guide. Thank you.

Let f(x)=x^3 −12x + 5

1. (a)  What is the y-intercept?

2. (b)  Find all critical points for the first derivative.

3. (c)  Use the second derivative test to determine whether these critical points are minima, maxima, or points of inflection.

4. (d)  Use the second derivative to find any points of inflection.

5. (e)  Sketch the graph, showing and labeling all extrema, points of inflection, and the y-intercept. You do not need to determine the x-intercepts.

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Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered

(a) y-intercept is the point where the graph crosses the y-axis i.e. when x=0

Plugging the value in the given function:

Hence, the y-intercept is 5

Step 2

(b) A point on a function is said to be a critical point if the value of the derivative at that point is 0

Applying derivative to the function,

A point on the given function is critical point if the value of the derivative is 0.

So,

Plugging the values in the function,

Therefore, the critical points are (2,-11) and (-2,21)

Step 3

(c)

As per the second derivative test, when a function has critical points at c and f(c)=0

If f ''(c)>0, then there is a local minimum at c

If f ''(c)<0, then there is a local maximum at c

If f ''(c)=0, then there is an inflection point at c

The second derivati...

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### Calculus 