Go with g[x] = Sin[0.5 x^2] - 0.5 and look at this graphic showing the plot of g

Question

Go with g[x] = Sin[0.5 x^2] - 0.5 and look at this graphic showing the plot of
g[x] and the plot of f[x] = Integrate[g[t], {t, 0, x}]
on the interval 0 x 4:

Discuss how you could have known in advance that f[x] (blue) would be going down when g[x](red) is negative, but that f[x] would be going up when g[x] is positive.

Make up your own function g[x] and your own choices of a and b and run the same experiment. Try to use a function g[x] that has both positive and negative values for different x's with a sxEs b. Describe what you see, paying special attention to what is doing when g[x] is positive and to what is doing when g[x] is negative What clue does this give you about the relationship between f[x] and g[x]?

Explanation / Answer

As g [x] is a curve like sinx curve only amplitude is different.

Now f [x] is the integration of g [x] so the curve is of cosx and substation of x/2.

So the value of f [x] increases when g [x] is positive but f [x] increases as we increase the value of x.

But when g [x] is -ve the value of f [x] is low but +ve.

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