How do you use the first and second derivatives to sketch #f(x)= x^4 - 2x^2 +3#?
The curve is concave downwards at
The curve is concave upwards at
The curve is concave upwards at
Given -
#y=x^4-2x^2+3#
#dy/dx=4x^3-4x#
#(d^2y)/(dx^2)=12x^2-4#
To sketch the graph, we have to find for what values of
#dy/dx=0 =>4x^3-4x=0#
#4x^3-4x=0#
#4x(x^2-1)=0#
#4x=0#
#x=0#
#x^2-1=0#
#x=+-sqrt1#
#x=1#
#x=-1#
The curve turns when At these points we have to decide whether the curve is concave upwards or concave downwards. For this we need the second derivatives - At Since the second derivative is less than zero, the curve is concave downwards at The value of the function is - The curve is concave downwards at At Since the second derivative is greater than zero, the curve is concave upwards at The value of the function is - The curve is concave upwards at At Since the second derivative is greater than zero, the curve is concave upwards at The value of the function is - The curve is concave upwards at
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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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