How do you find the critical points to sketch the graph #g(x)=x^4-8x^2-10#?
Start by finding the first derivative.
We now determine the second derivative to find the points of inflection.
Select test points between the critical points.
We will once again select test points.
It's true that you could just have found the next intervals of concavity and increasing/decreasing after the first by following the pattern of positive-negative-positive/negative-positive-negative, depending on the first interval, but I wanted to show you this method to make it as clear as possible.
The last thing I would like to discuss before graphing is intercepts. First, for the x-intercepts.
Finally, as for the y-intercept, we have:
We now trace the following graph putting all of the previous elements together.
graph{x^4 - 8x^2 - 10 [-58.5, 58.5, -29.27, 29.3]}
Hopefully this helps!
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To find the critical points of ( g(x) = x^4 - 8x^2 - 10 ), follow these steps:
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Compute the first derivative ( g'(x) ).
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Set ( g'(x) = 0 ) to find the values of ( x ).
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Solve for ( x ) to determine the critical points.
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Compute ( g'(x) ): [ g'(x) = 4x^3 - 16x ]
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Set ( g'(x) = 0 ): [ 4x^3 - 16x = 0 ]
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Factor out the common term: [ 4x(x^2 - 4) = 0 ]
Solve for ( x ): [ 4x = 0 \Rightarrow x = 0 ] [ x^2 - 4 = 0 \Rightarrow x = 2, x = -2 ]
The critical points are ( x = 0, x = 2, ) and ( x = -2 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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