Q.23: Verify Rolle's Theorem for the function f(x)=x2-x-12 in [-3,4]

Answer:

**Given the function is f(x)=x ^{2}-x-12
**
... (1)

(i) Putting x =-3 and x = 4, we get

f(-3) =
(-3)^{2}+3–12=12–12 = 0

and

f(4)= (4)^{2}
- 4 -12 = 16-16=0

Clearly f(-3) = f (4)

(ii) Since f (x) is polynomial function in x, then f (x) is continuous in [-3, 4].

(iii) Since f (x) is polynomial function in x, then it can be differentiate such that

f'(x)=2x-1

then by Rolle's theorem 3 at least c e(-3, 4) such that

f'(C)=0

2C-1=0

C=1/2

C=(1/2)
**
Î**
(-3,4) Hence verified Rolle's Theorem for [-3, 4].

** **

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