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March 26, 2017

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Find the intervals on which the f(x)= cos^2x is concave up and concave down.

  • calculus - ,

    To find if f(x) is concave up, we calculate the sign f"(x).
    The interval on which f"(x)>0 is concave up, and vice versa.

    Let f(x)=cos²(x),
    f'(x)=-2sin(x)cos(x)
    f"(x)=-2cos(2x)
    Now plot f"(x) between 0 and 2π to get:
    f"(x)≥0 for x∈[-π/4,π/4]
    f"(x)≤0 for x∈[π/4,3π/4]
    f"(x)≥0 for x∈[3π/4,5π/4]
    f"(x)≤0 for x∈[5π/4,7π/4]
    ...
    Can you take it from here?

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