Functions, Derivative of a function

[hard kernel]

Help

[hard kernel]

can someone help and explain these exercice please ?

Translation into English:

Determine (a), (b), and (c) such that the function (h(x)) is continuous and its derivative (h'(x)) is also continuous:

If your test scores are between 0 and 25, what does (h(x)) allow you to do?

Express all the constraints that make it possible to calculate the parameters (a), (b), and (c).

Deduce the matrix form that would allow these parameters to be calculated.

And

Translation into English:

1. Let (f(x) = -3x^2 + 6x - 2). For which value of (x) does this function reach its maximum?

2. The following functions, whose graphs you can see, are polynomials of degree 2, 3, and 4.
   For how many values do their derivatives become zero?

[mighty ruin]

alright so no1, here you need to check continuity of the function via left hand limits and right hand limits for x=10 and x=25

which gives you one equation

in a,b

then diffrentiate h(x) and do the continuity test again via left hand limit and right hand limits

which gives you another equation in a and b

then you can write those equation in matrix form and explain

[mighty ruin]

for this question
in 1) diffrentiate f(X) and find its critical point

this gives you its maxima

for a quadratic eq lika ax^2 + bx + c if a is >0 then ypu always get a minima whereas if a<0 you always get a maxima

now for 2)

as you can see you have polynomials of degree 2,3 and 4

if i you diffrentiate a poly with degree 2, you get a degree 1 eqn which gives only one point where their values are 0

for third, the diffrentiation gives us an eqn with degree 2, so 2 points