SOLUTION: Find the critical points of the following function. f(x)=2x^2-3x-5

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Question 1155422: Find the critical points of the following function.
f(x)=2x^2-3x-5
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Suppose that x=c is a critical point of f+%28x+%29.
If f'%28x+%29%3E0+to the left of x=c and f'%28x+%29%3C0 to the right of x=c, then x=c is a local maximum.
If f'%28x+%29%3C0+to the left of x=c and f'%28x+%29%3E0 to the right of x=c, then x=c is a local minimum.
If f'%28x+%29 is the same+sign on both sides of x=c then x=c is neither a local maximum nor a local minimum.

f%28x%29=2x%5E2-3x-5.....derivate
f'%28x%29=4x-3
4x-3=0
x=3%2F4-> a critical point: minimum
domain of f%28x%29 is
(-infinity,infinity)
f%28x%29 is monotone in intervals:
-infinity%3Cx%3C3%2F4 -> f%28x%29 is decreasing
3%2F4%3Cx%3Cinfinity->f%28x%29 is increasing


f%28x%29=2%283%2F4%29%5E2-3%283%2F4%29-5
f%28x%29=2%289%2F16%29-3%283%2F4%29-5
f%28x%29=%289%2F8%29-%289%2F4%29-5
f%28x%29=9%2F8-18%2F8-40%2F8
f%28x%29=-9%2F8-40%2F8
f%28x%29=-49%2F8
minimum is at: (3%2F4,-49%2F8)

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Find the critical points of the following function.
f(x)=2x^2-3x-5
There are 3 CRITICAL points of a parabola. They are:

1) the y-intercept (it's obvious that this is - 5)

2) the x-intercepts, if any [solving the equation for x, by setting f(x) = 0 will accomplish this task, and also, this equation can be FACTORED]

3) the VERTEX (this is found by determining the x-coordinate of the vertex, using the formula: matrix%281%2C3%2C+x%2C+%22=%22%2C+-+b%2F%282a%29%29, and then substituting this x-value into f(x)

   to get the y-coordinate of the vertex. These 2 coordinates will give you the vertex of the parabola, which by the way is a MINIMUM, as indicated by the

   leading coefficient being > 0. Alternatively, you can convert the function to the vertex form of a parabola, but that may be too much work!