To answer this question we need to remember what the derivative and the second derivative tells us geometrically:
The derivative of a function tells us the slope of the tangent line to the function; which means that we can determine if a function is increasing or decreasing if we look at the sign of its derivative:
• If the derivative is positive then the function is increasing.
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• If the derivative is negative then the function is decreasing.
The second derivative of a function tells us the concativity of a function:
• If the second derivative is positive then the function is concave up.
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• If the second derivative is negative then the function is concave dowm.
Now that we know this we can sketch a function:
For the first interval we know that the derivative is negative and the second derivative is also negative which means that the function has to be decreasing and concave down.
For the second interval we know that the derivative is negative and the second derivative is positve which means that the function has to be decreasing and concave up.