Calculus

Curve Sketching

Increasing / Decreasing Functions

         

What is the range of all possible values aa such that f(x)=x3ax2+(a+6)x+1f(x) = x^3-ax^2+(a+6)x+1 is an everywhere increasing function?

What is the range of possible values aa such that the function f(x)=x3+ax218x1f(x) = -x^3+ax^2-18x-1 is an always decreasing function?

The above diagram shows the curve y=f(x).y=f(x). Let aa be the xx-coordinate of the intersection point between the curve and negative part of the xx-axis, and let bb and cc be the xx-coordinates of the local maximum and minimum of the curve, respectively, as shown in the diagram. If a=3,b=4,c=7,a = -3, b = 4, c = 7, and f(x)>0f'(x)>0 for the portions of the curve that are not displayed in the diagram, what is the range of xx that satisfies f(x)f(x)>0?f(x)f'(x) > 0?

What is the range of possible values of aa such that the function f(x)=3x44(a+7)x3+6(7a+6)x272ax+1f(x)=3x^4 - 4(a+7)x^3+6(7a+6)x^2-72ax+1 increases in the interval x6?x \geq 6?

What is the sum of the minimum and maximum values of the constant aa such that f(x)=13x3+ax2+(14a40)x+12f(x) = \frac{1}{3}x^3 + ax^2 + (14 a - 40)x + 12 is an increasing function in the interval (, )? (-\infty,\ \infty)?

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