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Q:

 A curve with equation \(y=ax^{3} +bx\)  passes through the origin at an angle of  \(45° \)

Passes through the point \(( 2 , - 6 ) \)

a) Find the values of \(a\) and \(b\).

Q:

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

\(\text{f}(x)=5x^2-2x+1,~[0,~2]\)

 

a. Yes, it does not matter If \(\mathrm{f}\) is continuous or differentiable, every function satisfies the Mean Value Theorem.

b. Yes, \(\mathrm{f}\) is continuous on \([0, 2]\) and differentiable on \((0, 2)\) since polynomials are continuous and differentiable on \(\mathbb{R}\).

c. No, \(\mathrm{f}\) is not continuous on \([0, 2]\).

d. No, \(\mathrm{f}\) is continuous on \([0, 2]\) but not differentiable on \((0, 2)\).

e. There is not enough information to verify if this function satisfies the Mean Value Theorem.

 

If it satisfies the hypotheses, find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem.