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chjopl
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Find the local maxima:
f(x)=2x^3 - 3x^2 -12x + 3
f(x)=2x^3 - 3x^2 -12x + 3
Taking the derivative with respect to x, we get 6x^2 - 6x - 12. Simplified to 6(x^2 - x + 2). That there's a parabola facing up, so it doesn't have a local max, but does have an absolute min.chjopl said:Find the local maxima:
f(x)=2x^3 - 3x^2 -12x + 3
Same deal. The derivative is 1 + 1/x, which is 1/x shifted up one unit. Goes off into infinity, as well.chjopl said:does a local max exist for f(x)=x + ln(x)
Planckenstein said:Same deal. The derivative is 1 + 1/x, which is 1/x shifted up one unit. Goes off into infinity, as well.
Unless these have a bounded domain, they both don't have local maxes.
Planckenstein said:Taking the derivative with respect to x, we get 6x^2 - 6x - 12. Simplified to 6(x^2 - x + 2). That there's a parabola facing up, so it doesn't have a local max, but does have an absolute min.
Kreil said:Just because the derivative has 2 zeroes doesn't mean the function has to have a max and a min...it could pass through the x-axis at 1 point (the zero) and come back up, never becoming negative.
In order for their to be a max or min at a certain point, the zeroes of the first derivative are the places it would happen.
kreil said:What I said does not convey the idea I was trying to convey
Imagine the graph of a first derivative...it is positive and decreasing, touches the x-axis at 1 point (does NOT pass through the x axis), then increases. It's values over this interval are never negative. The graph of f changed concavity but does NOT have a local extremum at this point since the first derivative did not change signs.
That is what I was trying to say. Not every zero of the first derivative is guaranteed to be an extremum.
A local maximum is a point on a graph where the function reaches its highest value within a specific interval, but may not be the overall highest point on the graph. It is also known as a relative maximum.
To find local maxima graphically, you can plot the function on a graph and visually identify the highest points on the graph within a specific interval. You can also use a graphing calculator or software to plot the function and find the coordinates of the local maxima.
To find the local maxima algebraically, you can take the derivative of the function and set it equal to zero. Then, solve for the values of x that make the derivative equal to zero. These values will be the critical points of the function, and you can plug them back into the original function to find the corresponding y-values, which will be the local maxima.
Yes, a function can have more than one local maximum. This occurs when the function has multiple peaks within a specific interval. These points will have the same x-value but different y-values.
The local maximum of f(x)=2x^3-3x^2-12x+3 is (1, -10). This is found by taking the derivative of the function, setting it equal to zero, solving for x, and plugging the value back into the original function to find the corresponding y-value.