Explain why we placed the value of x= 4/3( the minimum value) into the equ of gradient(dy/dx) [in the answer, marking scheme attached] to get the answer -4/3 ? ? isn't 4/3 the minimum

are the critical points, and judging by the picture alone, you must have and . (You might want to verify with the derivative test in case that's expected.)

Then the shaded region has area



I'll leave the details to you.

Now, for part (iv), you're asked to find the minimum of , which entails first finding the second derivative:




setting equal to 0 and finding the critical point:



This is to say the minimum value of *occurs when *, but this is not necessarily the same as saying that is the actual minimum value.

The minimum value of is obtained by evaluating the derivative at this critical point:

3x+9+4x+x=8x+9that's the answer for your question.