# For what values of x is #f(x)= -9x^3 + 4 x^2 + 7x -2 # concave or convex?

by the power rule, and so the derivative of this is

Here's a graph of the function, with the inflection point circled. graph{(-9x^3+4x^2+7x-2-y)((x-4/27)^2+(y+0.9)^2/36-0.0025)=0 [-2.1, 2.1, -6, 6]}

Notice how the function is V-shaped to the left of the inflection point, and cave-shaped to its right. That's an easy way to remember the difference between concave and convex: concave is like a cave; convex is like a V.

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We calculate the first derivative and build a chart of variations

To determine the critical points, we solve the equation

The chart of variations is

So,

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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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