Calculus

Extrema

Curve Sketching

         

Let PP and QQ be the two inflection points of the curve y=e5x2.\displaystyle y=e^{-5x^2}. What is the area of the triangle OPQ,OPQ, where OO is the origin?

If f(x)=1x2+9,\displaystyle{f(x)=\frac{1}{x^2+9}}, the interval on which the curve y=f(x)y=f(x) is concave down can be expressed as a<x<b.a<x<b. What is the value of ab?ab?

Let f(x)=8+ax+bx2,\displaystyle{f(x)=8+\frac{a}{x}+\frac{b}{x^2},} where aa and bb are constants. If the point (1,0)(-1,0) is the inflection point of the curve y=f(x),y=f(x), what is a+b?a+b?

If f(x)=x1/3(x+26) f(x) = x^{1/3}(x + 26) , the interval on which the curve y=f(x) y = f(x) is concave down can be expressed as a<x<b a < x < b . What is the value of a+b a + b ?

Let y=(ax2+30)exy=(ax^2+30)e^x be the equation of a curve, where aa is a constant. If AA and BB are two distinct arbitrary points on the curve and the arc between the two points always lies below the line connecting the two points, what is the range of a?a?

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