How do you find the intervals of increasing and decreasing using the first derivative given #y=-2x^2+4x+3#?

Answer 1

#f(x)# is increasing from #(-oo,1)#
#f(x)# is decreasing from #(1,oo)#

We want to perform that first derivative test here:
We begin by differentiate using the power rule:

#d/dxx^n=nx^(n-1)#

#d/dx=-2(2)x^(2-1)+4(1)x^cancel(1-1)+0#

Keep in mind that #x^0=1# and that derivative of a constant is zero.

#f'(x)=-4x+4#

Now we want to factor and set it equal to zero:

#-4(x-1)=0#

#x-1=0#

#x=1#

We create a test a interval from #(-oo,1)uu(1,oo)#
Now you pick numbers in between the interval and test them in the derivative. If the number is positive this means the function is increasing and if it's negative the function is decreasing.

I picked 0 a number from the left

#f'(0)=4#

This means from #(oo,1)# the function is increasing.

Then I picked a number from the right which was 2.

#f'(2)=-4#

This means from #(-1,oo)# the function is decreasing.

So, from #(oo,1)# the function is increasing and from #(-1,oo)# the function is decreasing.

Note: For this exact reason we can say that there's an absolute max at #f(1)#. We can say this because its only a parabola.

Attach is an image that may help you:

The graph will help you visualize it better.

graph{-2x^2+4x+3 [-10, 10, -5, 5]}

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Answer 2
To find the intervals of increasing and decreasing using the first derivative, we first need to find the first derivative of the given function \( y = -2x^2 + 4x + 3 \), which we'll denote as \( f(x) \). \[ f'(x) = \frac{d}{dx}(-2x^2 + 4x + 3) \] \[ f'(x) = -4x + 4 \] Now, to determine the intervals of increasing and decreasing, we need to analyze where the first derivative is positive (indicating increasing) and where it is negative (indicating decreasing). For \( f'(x) = -4x + 4 \): - The function is increasing when \( f'(x) > 0 \). - The function is decreasing when \( f'(x) < 0 \). To find where \( f'(x) > 0 \): \[ -4x + 4 > 0 \] \[ -4x > -4 \] \[ x < 1 \] To find where \( f'(x) < 0 \): \[ -4x + 4 < 0 \] \[ -4x < -4 \] \[ x > 1 \] Therefore, the function is increasing for \( x < 1 \) and decreasing for \( x > 1 \). These intervals represent the intervals of increasing and decreasing, respectively.
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Answer from HIX Tutor

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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