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A turning point occurs when the velocity is equal to zero, but the acceleration is not equal to zero.
t(x)=(x+5)^3+7
dt/dx=3(x+5)^2
d2t/dx2=6(x-5)
dt/dx=0 only when x=-5
However, since d2t/dx2(-5)=0, this point is an inflection point, not a turning point.
So there is no turning point for this function.
Now in this problem, it is even easier than the above to show that there is no turning point. A turning point by definition is when the derivative or velocity changes sign. Since in this case v=3(x+5)^2, for any value of x, v≥0, and thus never becomes negative, so it never changes from a positive to negative velocity because velocity in this instance is a squared function.
-5x - 15 > 10 + 20x...subtract 10 from both sides, add 5x to both sides
-10 - 15 > 20x + 5x ...simplify
-25 > 25x...divide both sides by 25
-25/25 > x ....reduce
-1 > x or x < -1 <===
It depends on how much gas is for 1 gallon
